This article is part of the series Proceedings of the International Congress in Honour of Professor Hari M. Srivastava.

Open Access Research

On residual algebraic torsion extensions of a valuation of a field K to K ( x 1 , , x n )

Figen Öke

Author Affiliations

Department of Mathematics, Faculty of Science, Trakya University, Balkan Campus, Edirne, 22030, Turkey

Fixed Point Theory and Applications 2013, 2013:46  doi:10.1186/1687-1812-2013-46


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2013/1/46


Received:11 December 2012
Accepted:14 February 2013
Published:5 March 2013

© 2013 Öke; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let v be a valuation of a field K with a value group <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M2">View MathML</a> and a residue field <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M3">View MathML</a>, w be an extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4">View MathML</a>. Then w is called a residual algebraic torsion extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M6">View MathML</a> is an algebraic extension and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M7">View MathML</a> is a torsion group. In this paper, a residual algebraic torsion extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1">View MathML</a> is described and its certain properties are investigated. Also, the existence of a residual algebraic torsion extension of a valuation on K to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1">View MathML</a> with given residue field and value group is studied.

MSC: 12J10, 12J20, 12F20.

Keywords:
extensions of valuations; residual algebraic torsion extensions; valued fields; value group; residue field

1 Introduction

Let K be a field, v be a valuation on K with a value group <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M2">View MathML</a> and a residue field <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M3">View MathML</a>. The big target is to define all extensions of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1">View MathML</a>. Residual transcendental extensions of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4">View MathML</a> are described by Popescu, Alexandru and Zaharescu in [1,2]. Residual algebraic torsion extensions of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4">View MathML</a> are studied for the first time in [3]. A residual transcendental extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1">View MathML</a> is defined in [4] by Öke. These studies are summarized in the second section. The paper is aimed to study residual algebraic torsion extensions of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1">View MathML</a>. In the third section, a residual algebraic torsion extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1">View MathML</a> is defined and certain properties of the residual algebraic torsion extensions given in [3] are generalized. In the last section, the existence of an r.a.t. extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1">View MathML</a> with given residue field and value group is demonstrated.

2 Preliminaries and some notations

Throughout this paper, v is a valuation of a field K with a value group <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M2">View MathML</a>, a valuation ring <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M20">View MathML</a> and a residue field <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M21">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M22">View MathML</a> is an algebraic closure of K, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M23">View MathML</a> is a fixed extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M22">View MathML</a>. The value group of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M23">View MathML</a> is the divisible closure of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M2">View MathML</a> and its residue field is the algebraic closure of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M21">View MathML</a>. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1">View MathML</a> are rational function fields over K with one and n variables respectively. For any α in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M20">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M31">View MathML</a> denotes its natural image in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M3">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M33">View MathML</a>, then the restriction of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M23">View MathML</a> to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M35">View MathML</a> will be denoted by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M36">View MathML</a>.

Let w be an extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4">View MathML</a>. Then w is called a residual transcendental (r.t.) extension of v if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M38">View MathML</a> is a transcendental extension.

The valuation w, which is defined for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M39">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M40">View MathML</a> is called Gauss extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4">View MathML</a>, its residue field is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M42">View MathML</a>, is the simple transcendental extension of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M3">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M44">View MathML</a>[5].

The valuation <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M45">View MathML</a>, which is defined for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M46">View MathML</a> as

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M47">View MathML</a>

(1)

is called a valuation defined by the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M48">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M49">View MathML</a> is called a pair of definitions of w. Also, w is an r.t. extension of v. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M50">View MathML</a> for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M51">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M52">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M53">View MathML</a> is called a minimal pair with respect to K[2].

If w is an r.t. extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4">View MathML</a>, there exists a minimal pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M55">View MathML</a> such that a is separable over K. Two pairs <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M56">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M57">View MathML</a> define the same valuation w if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M58">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M59">View MathML</a>[2]. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M60">View MathML</a> be the minimal polynomial of a with respect to K and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M61">View MathML</a>. For each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M62">View MathML</a>, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M63">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M64">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M65">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M66">View MathML</a>, be the f-expansion of F. Then w is defined as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M67">View MathML</a>

(2)

Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M68">View MathML</a>. Let e be the smallest non-zero positive integer such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M69">View MathML</a>. Then there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M70">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M71">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M72">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M73">View MathML</a> is an element of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M74">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M75">View MathML</a> is transcendental over <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M3">View MathML</a>. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M77">View MathML</a> can be identified canonically with the algebraic closure of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M3">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M79">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M80">View MathML</a>[2].

Let w be an extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4">View MathML</a>. w is called a residual algebraic (r.a.) extension of v if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M38">View MathML</a> is an algebraic extension. If w is an r.a. extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M84">View MathML</a> is not a torsion group, then w is called a residual algebraic free (r.a.f.) extension of v. In this case, the quotient group <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M85">View MathML</a> is a free abelian group. More precisely, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M85">View MathML</a> is isomorphic to Z[3].

w is called a residual algebraic torsion (r.a.t) extension of v if w is an r.a. extension of v and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M87">View MathML</a> is a torsion group. In this case, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M88">View MathML</a> is satisfied [3].

The order relation on the set of all r.t. extensions of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4">View MathML</a> is defined as follows: <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M90">View MathML</a> for all polynomials <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M91">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M92">View MathML</a> and there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M91">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M94">View MathML</a>, then it is written <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M95">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M96">View MathML</a> be minimal pairs of the definition of the r.t. extensions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M97">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M98">View MathML</a> of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4">View MathML</a>, respectively. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M100">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M101">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M102">View MathML</a>; moreover, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M95">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M101">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M105">View MathML</a>[3].

Let I be a well-ordered set without the last element and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M106">View MathML</a> be an ordered system of r.t. extensions of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M108">View MathML</a> is defined by a minimal pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M109">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M110">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M111">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M112">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M113">View MathML</a> is called an ordered system of r.t. extensions of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4">View MathML</a>.

Then the valuation of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4">View MathML</a> defined as

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M116">View MathML</a>

(3)

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M91">View MathML</a> is an extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4">View MathML</a> and it is called a limit of the ordered system <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M113">View MathML</a>. w may not be an r.t. extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M4">View MathML</a>[3].

Using the above studies an r.a.t extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1">View MathML</a> can be defined. For this reason the r.t. extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1">View MathML</a> defined in [4] can be used. An r.t. extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1">View MathML</a> is defined by using r.t. extensions of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M124">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M125">View MathML</a> in [4].

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M126">View MathML</a> be an r.t. extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M127">View MathML</a> defined by a minimal pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M128">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M129">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M130">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M131">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M132">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M125">View MathML</a>. Each polynomial <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M134">View MathML</a> can be uniquely written as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M135">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M136">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M137">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M129">View MathML</a>.

The valuation w defined as

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M139">View MathML</a>

(4)

is an extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1">View MathML</a>. u is an r.t. extension of v which is a common extension of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M141">View MathML</a> to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M142">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M143">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M144">View MathML</a> be the smallest positive integer such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M145">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M146">View MathML</a> is the restriction of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M147">View MathML</a> to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M148">View MathML</a>. Then there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M149">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M150">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M151">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M152">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M153">View MathML</a> is transcendental over <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M21">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M155">View MathML</a>. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M156">View MathML</a> can be canonically identified with the algebraic closure of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M21">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M158">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M159">View MathML</a>[4].

In the next section, an r.a.t extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1">View MathML</a> will be defined by using that r.t. extension.

3 A residual algebraic torsion extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1">View MathML</a>

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M126">View MathML</a> be an r.t. extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M127">View MathML</a> defined by a minimal pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M128">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M125">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M130">View MathML</a> and let u be the r.t. extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M142">View MathML</a> defined as in (4). Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M168">View MathML</a> be an r.t. extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M127">View MathML</a> defined by a minimal pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M170">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M125">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M172">View MathML</a> and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M173">View MathML</a> be the r.t. extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174">View MathML</a> defined as in (4). A relation between such kind of r.t. extensions of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174">View MathML</a> can be defined so that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M176">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M177">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M125">View MathML</a>. This is an order relation, and if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M176">View MathML</a>, then for each polynomial <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M180">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M181">View MathML</a> is satisfied. Because, for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M182">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M183">View MathML</a>

Now, let I be a well-ordered set without the last element and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M184">View MathML</a> be an ordered system of r.t. extensions of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M108">View MathML</a> is defined as in (4), i.e., <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M187">View MathML</a> is the common extension of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M188">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M188">View MathML</a> is the r.t. extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M124">View MathML</a> defined by the minimal pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M191">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M192">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M110">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M111">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M112">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M113">View MathML</a> is an ordered system of r.t. extensions of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1">View MathML</a>. Then the valuation w of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1">View MathML</a> defined as

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M199">View MathML</a>

(5)

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M180">View MathML</a> is an extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1">View MathML</a> and it is called a limit of the ordered system <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M113">View MathML</a>.

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M203">View MathML</a> is the restriction of w to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M124">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M125">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M203">View MathML</a> is the limit of the ordered system <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M207">View MathML</a> of r.t. extensions of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M124">View MathML</a>. Also, w is the common extension of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M209">View MathML</a> to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M203">View MathML</a> may not be an r.t. extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M124">View MathML</a>, then w may not be an r.t. extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1">View MathML</a>.

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M214">View MathML</a> is a residual algebraic torsion extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M1">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M216">View MathML</a> is satisfied. Some other properties of w are studied below.

Denote the extension of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M187">View MathML</a> to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M218">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M219">View MathML</a> and the extension of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M220">View MathML</a> to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M221">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M222">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M155">View MathML</a> and for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M110">View MathML</a>.

Theorem 3.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M225">View MathML</a>be an ordered system of r.t. extensions of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M147">View MathML</a>to<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M218">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M219">View MathML</a>is defined as in (4), i.e., <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M108">View MathML</a>is the r.t. extension ofvto<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M218">View MathML</a>which is the common extension of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M231">View MathML</a>for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M125">View MathML</a>and for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M233">View MathML</a>. Denote the restriction of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M234">View MathML</a>to<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M142">View MathML</a>by<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M187">View MathML</a>and the restriction of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M23">View MathML</a>to<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M238">View MathML</a>by<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M239">View MathML</a>. Then

1. For all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M240">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M112">View MathML</a>, one has<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M242">View MathML</a>, i.e., <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M113">View MathML</a>is an ordered system of r.t. extensions ofvto<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174">View MathML</a>.

2. For all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M245">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M112">View MathML</a>, one has<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M247">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M248">View MathML</a>.

3. Suppose that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M249">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M250">View MathML</a>is not an r.t. extension of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M23">View MathML</a>to<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M218">View MathML</a>and denote thatwis the restriction of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M250">View MathML</a>to<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174">View MathML</a>. Then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M255">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M256">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M257">View MathML</a>.

Proof For every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M110">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M155">View MathML</a>, denote that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M260">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M261">View MathML</a>.

1. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M262">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M240">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M112">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M265">View MathML</a>. We show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M242">View MathML</a>. Assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M267">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M187">View MathML</a> is the common extension of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M231">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M270">View MathML</a> is the common extension of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M271">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M125">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M273">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M125">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M275">View MathML</a> is a minimal pair of the definition of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M220">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M277">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M125">View MathML</a>. But it is a contradiction, because <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M279">View MathML</a> is an ordered system of r.t. extensions of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M147">View MathML</a> to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M281">View MathML</a> and so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M282">View MathML</a>, i.e., <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M283">View MathML</a>[3]. Hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M284">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M125">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M187">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M270">View MathML</a> are common extensions of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M288">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M289">View MathML</a> respectively for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M125">View MathML</a> and for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M110">View MathML</a>, it is concluded that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M242">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M293">View MathML</a>.

2. It is enough to study for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M294">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M295">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M296">View MathML</a>.

It is seen that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M297">View MathML</a> by using the [[3], Th. 2.3] and this gives <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M298">View MathML</a>.

Assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M299">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M300">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M275">View MathML</a> is a minimal pair of the definition of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M188">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M303">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M304">View MathML</a> coincides with the <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M305">View MathML</a> which is the residue of B in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M306">View MathML</a>. Hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M307">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M245">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M112">View MathML</a>.

3. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M310">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M311">View MathML</a> and w is not an r.t. extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M142">View MathML</a>. Using [[3], Th. 2.3] and the definition of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M188">View MathML</a>, the proof can be completed. Take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M295">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M315">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M275">View MathML</a> is a minimal pair of the definition of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M231">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M318">View MathML</a>. This means that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M319">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M110">View MathML</a> and so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M321">View MathML</a>.

Conversely, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M322">View MathML</a> be the restriction of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M219">View MathML</a> to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M324">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M155">View MathML</a> and for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M110">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M327">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M328">View MathML</a> for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M329">View MathML</a>. This gives <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M330">View MathML</a>.

Now, assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M331">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M332">View MathML</a> and since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M333">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M334">View MathML</a>, which is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M335">View MathML</a>-residue of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M336">View MathML</a>, coincides with the residue of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M336">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M158">View MathML</a>. This shows <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M339">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M110">View MathML</a>, and then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M341">View MathML</a> .

For the reverse inclusion, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M342">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M343">View MathML</a>. For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M155">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M345">View MathML</a> is equal to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M346">View MathML</a> and so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M347">View MathML</a>. Hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M348">View MathML</a>.

 □

The following theorem can be obtained as a result of Theorem 3.1.

Corollary 3.2Under the above notations, letwbe an r.a.t. extension ofvto<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174">View MathML</a>. Then the following are satisfied:

1. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M248">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M351">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M240">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M112">View MathML</a>.

2. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M113">View MathML</a>is an ordered system of r.t. extensions ofvto<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M356">View MathML</a>. Moreover, we have<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M357">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M358">View MathML</a>.

Proof If w is an r.a.t. extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M250">View MathML</a> is an r.a.t. extension <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M147">View MathML</a> to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M362">View MathML</a> and so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M363">View MathML</a> is an r.a.t. extension of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M147">View MathML</a> to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M281">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M125">View MathML</a>. We can take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M367">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M155">View MathML</a> as co-final well-ordered subsets of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M369">View MathML</a>. I has no last element because <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M363">View MathML</a> is not an r.t. extension of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M23">View MathML</a>. For every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M110">View MathML</a>, choose the element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M373">View MathML</a> such that for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M155">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M375">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M376">View MathML</a> is the smallest possible for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M377">View MathML</a>. This means that if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M378">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M379">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M380">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M275">View MathML</a> is a minimal pair of the definition of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M222">View MathML</a> with respect to K for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M155">View MathML</a>. According to [[3], Th. 4.1], <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M384">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M112">View MathML</a>, which means that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M386">View MathML</a> is an ordered system of r.t. extensions of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M147">View MathML</a> to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M388">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M155">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M386">View MathML</a> has a limit <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M391">View MathML</a> which is an r.a.t extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M388">View MathML</a>. For all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M110">View MathML</a>, take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M219">View MathML</a> as the common extension of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M222">View MathML</a> to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M250">View MathML</a> as the common extension of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M398">View MathML</a> to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M218">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M155">View MathML</a>. Denote the restriction of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M234">View MathML</a> to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M402">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M187">View MathML</a> and denote the restriction of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M250">View MathML</a> to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174">View MathML</a> by w. In the same way as that in the proof of Theorem 3.1, it is seen that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M242">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M407">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M112">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M409">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M110">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M255">View MathML</a>. Moreover, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M412">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M257">View MathML</a> are satisfied. □

4 Existence of r.a.t. extensions of valuations of K to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174">View MathML</a> with given residue field and value group

It can be concluded from section three and from [3] that if w is an r.a.t. extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M6">View MathML</a> is a countable generated infinite algebraic extension and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M7">View MathML</a> is a countable infinite torsion group. In this section, the converse is studied.

Theorem 4.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M418">View MathML</a>be a countably generated infinite algebraic extension andGbe an ordered group such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M419">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M420">View MathML</a>is a countably infinite torsion group. Then there exists an r.a.t. extensionwofvto<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M422">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M423">View MathML</a>.

Proof Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M424">View MathML</a> is the algebraic closure of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M3">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M426">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M418">View MathML</a> is countably generated, there exists a tower of fields <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M428">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M429">View MathML</a>, and since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M420">View MathML</a> is a countable torsion group, there exists a sequence of subgroups of G such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M431">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M432">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M433">View MathML</a> is finite for all s and that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M434">View MathML</a>. According to [[6], Th. 3.2], there exists an r.t. extension <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M435">View MathML</a> of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M437">View MathML</a>, the algebraic closure of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M3">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M439">View MathML</a> is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M440">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M441">View MathML</a> and if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M442">View MathML</a>, then the restriction of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M435">View MathML</a> to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M444">View MathML</a> is not the Gauss extension of the restriction of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M435">View MathML</a> to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M127">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M447">View MathML</a> and for all s. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M448">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M449">View MathML</a> is transcendental over <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M440">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M451">View MathML</a> and for all s. Denote the restriction of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M435">View MathML</a> to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M453">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M454">View MathML</a> and the algebraic closure of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M3">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M456">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M457">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M155">View MathML</a> and for all s. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M459">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M449">View MathML</a> is transcendental over <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M461">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M303">View MathML</a> and for all s.

Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M463">View MathML</a> is the tower of finite extensions of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M3">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M155">View MathML</a>. Denote <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M466">View MathML</a>. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M467">View MathML</a> is the sequence of subgroups of G such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M468">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M469">View MathML</a> is finite for all s and for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M125">View MathML</a>. Then there exists an r.a.t. extension <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M203">View MathML</a> of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M124">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M473">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M474">View MathML</a>[3].

It means that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M475">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M476">View MathML</a> are algebraic independent over K, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M477">View MathML</a> is a countable generated infinite algebraic extension and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M478">View MathML</a> is a countable torsion group. Hence there exists an r.a.t. extension <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M479">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M480">View MathML</a> to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M481">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M482">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M483">View MathML</a>. Using the induction on n, it is obtained that there exits an r.a.t. extension <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M484">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M485">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M486">View MathML</a> to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M488">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M489">View MathML</a>

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M335">View MathML</a> is an r.a.t. extension of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M491">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M492">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M493">View MathML</a> is an r.a.t. extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174">View MathML</a>. □

Theorem 4.2Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M495">View MathML</a>be a finite extension, Gbe an ordered group such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M419">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M497">View MathML</a>is finite. Assume that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M498">View MathML</a>. Then there exists an r.a.t. extension ofvto<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M500">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M423">View MathML</a>.

Proof Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M495">View MathML</a> is a finite extension, it can be written that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M503">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M504">View MathML</a> is algebraic over <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M21">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M506">View MathML</a>. It can be taken <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M507">View MathML</a>, because if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M508">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M509">View MathML</a> elements can be chosen as equal. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M420">View MathML</a> is finite, there exists a sequence of subgroups of G such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M511">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M512">View MathML</a> is finite for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M513">View MathML</a>.

Hence there exists an r.a.t. extension <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M97">View MathML</a> of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M515">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M516">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M517">View MathML</a>[3]. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M518">View MathML</a> be the completion of K with respect to v and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M519">View MathML</a> be the extension of v to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M518">View MathML</a>. According to [[7], Prop. 1], the completion of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M515">View MathML</a> with respect to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M522">View MathML</a> is isomorphic to a field belonging to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M523">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M524">View MathML</a> is the completion of the algebraic closure Ω of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M518">View MathML</a> with respect to the unique extension of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M526">View MathML</a> to Ω and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M523">View MathML</a> is the set of complete fields L such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M528">View MathML</a>. Moreover, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M498">View MathML</a>, there exists an element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M530','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M530">View MathML</a> which is transcendental over K. That is, there exists a Cauchy sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M531','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M531">View MathML</a> which converges to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M532">View MathML</a>.

Therefore if we denote the completion of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M515">View MathML</a> with respect to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M522">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M535','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M535">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M536">View MathML</a>. Also, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M537','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M537">View MathML</a> is finite, then there exists an r.a.t. extension <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M538','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M538">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M522">View MathML</a> to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M481">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M541','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M541">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M542','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M542">View MathML</a>. Using the induction, it is obtained that there exists an r.a.t. extension <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M543">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M544','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M544">View MathML</a> on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M545','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M545">View MathML</a> to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M546','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M546">View MathML</a> such that its residue field is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M547','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M547">View MathML</a> and its value group is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M548','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M548">View MathML</a>. Finally, there exists an r.a.t. extension <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M549','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M549">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M543">View MathML</a> to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M174">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M552','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M552">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/46/mathml/M423">View MathML</a>. □

Competing interests

The authors declare that they have no competing interests.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

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