SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

Analysis approach to finite monoids

A Sinan Çevik1*, I Naci Cangül2 and Yılmaz Şimşek3

Author Affiliations

1 Department of Mathematics, Faculty of Science, Selçuk University, Campus, Konya, 42075, Turkey

2 Department of Mathematics, Faculty of Arts and Science, Uludag University, Gorukle Campus, Bursa, 16059, Turkey

3 Department of Mathematics, Faculty of Art and Science, Akdeniz University, Campus, Antalya, 07058, Turkey

For all author emails, please log on.

Fixed Point Theory and Applications 2013, 2013:15  doi:10.1186/1687-1812-2013-15


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2013/1/15


Received:19 November 2012
Accepted:9 January 2013
Published:22 January 2013

© 2013 Çevik et al.; 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

In a previous paper by the authors, a new approach between algebra and analysis has been recently developed. In detail, it has been generally described how one can express some algebraic properties in terms of special generating functions. To continue the study of this approach, in here, we state and prove that the presentation which has the minimal number of generators of the split extension of two finite monogenic monoids has different sets of generating functions (such that the number of these functions is equal to the number of generators) that represent the exponent sums of the generating pictures of this presentation. This study can be thought of as a mixture of pure analysis, topology and geometry within the purposes of this journal.

AMS Subject Classification: 11B68, 11S40, 12D10, 20M05, 20M50, 26C05, 26C10.

Keywords:
efficiency; p-Cockcroft property; split extension; generating functions; Stirling numbers; array polynomials

1 Introduction and preliminaries

Associated with any (connected) topological space X is its fundamental group <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M1">View MathML</a> or 2-complex (Squier complex) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M2">View MathML</a>. This can often be specified by means of a presentation. A presentation of a group G or monoid M consists of a set of generators of G or M, together with a collection of relations amongst these generators, such that any other relation amongst the generators is derivable (in a precise sense) from the given relations. Algebraic information about <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M1">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M2">View MathML</a> can be used to obtain topological information about X (cf.[1]). Many techniques of this branch of mathematics are purely algebraic, and it is possible to achieve much using these techniques. However, in recent years many techniques involving geometric ideas have emerged and are proving more fruitful. These geometric techniques involve graph theory, the theory of tessellations of various surfaces and covering space theory, to name a few.

The number of vertex-colorings of a graph is given by a polynomial on the number of used colors (see [2]). Based on this polynomial, one can define the chromatic number as the minimum number of colors such that the chromatic polynomial is positive. Recently, our attention has been drawn to the paper [3] which is a generalization on the chromatic polynomial of a graph subdivision, and basically the authors determine the chromatic number for a simple graph and then present the generalized polynomial for a particular case of graph subdivision. In this reference, the main idea was to express some graph theoretical parameters in terms of special functions. In a similar manner within algebra, by considering a group or a monoid presentation <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M5">View MathML</a>, an approximation from algebra to analysis has been recently developed [4]. To do that, the authors supposed <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M5">View MathML</a> satisfies the special algebraic properties either efficiency or inefficiency while it is minimal. (The reason for choosing efficiency or (minimal) inefficiency was to have an advantage to work on a minimal number of generators.) Then it was investigated whether some generating functions can be applied, and then it was studied what kind of new properties can be obtained by considering special generating functions over <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M5">View MathML</a>. In fact, to investigate this theory, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M5">View MathML</a> has been taken as the presentation of the split extension <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M9">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M10">View MathML</a>, respectively. Since the results in [3] imply a new studying area for graphs in the meaning of representation of parameters by generating functions, the results in [4] will be also given an opportunity to make a new classification of infinite groups and monoids by using generating functions.

This paper can be thought of as another version of [4]. Our general aim here is to define some generating functions in terms of the minimality of the given presentation. This will imply that the minimal number of generators can be represented as generating functions. Similarly as in [4], our approximation will be applied by considering the split extension. Here, the split extension will be defined as a semi-direct product of two finite monogenic (cyclic) monoids (we may refer to [5] for details on these monoids). It is obvious that the split extension of two finite structures will also be finite. So, the main difference between the results in here and in the paper [4] lies in this fact. Because, while a classification over special cases of infinite groups or monoids was given in [4], the classification in the present paper only focuses on the finite monoids. It is well known that giving some different approximations over finite cases is also as important as giving those over infinite cases.

In the following first subsection, as supportive material, some algebraic facts over split extensions (equivalently, semi-direct products), presentations of finite monogenic monoids, a trivializer set of these presentations and efficiency (equivalently, p-Cockcroft property) are reminded. In Section 2, we present the main material of this paper as two separate subsections. In the first subsection, we present some known results about necessary conditions for the presentation, say <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M11">View MathML</a>, of the split extension of two finite monogenic monoids to be p-Cockcroft (see Proposition 2.1 below) and to be minimal but inefficient (see Proposition 2.3 below). In the final subsection, as a result of all theories until there, we introduce generating functions related to our title (see Theorems 2.5, 2.7 and 2.12 below). In Section 3, by considering one of the functions defined in the previous section, we study this function in the meaning of again generating functions and functional equations (see Theorems 3.1 and 3.3 below).

1.1 Fundamentals of the algebraic part

This subsection should be completely thought of as a part of the expressions in the beginning of this paper.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M12">View MathML</a> be a monoid presentation where a typical element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M13">View MathML</a> has the form <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M14">View MathML</a>. Here <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M15">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M16">View MathML</a> are words on X (that is, elements of the free monoid <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M17">View MathML</a> on X). The monoid defined by<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M18">View MathML</a> is the quotient of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M17">View MathML</a> by the smallest congruence generated by r.

We have a (Squier) graph <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M20">View MathML</a> associated with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M18">View MathML</a>, where the vertices are the elements of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M17">View MathML</a> and the edges are the 4-tuples <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M23">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M24">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M25">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M26">View MathML</a>. The initial, terminal and inversion functions for an edge e as given above are defined by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M27">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M28">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M29">View MathML</a>.

Two paths π and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M30">View MathML</a> in a 2-complex are equivalent if there is a finite sequence of paths <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M31">View MathML</a>, where for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M32">View MathML</a>, the path <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M33">View MathML</a> is obtained from <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M34">View MathML</a> either by inserting or deleting a pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M35">View MathML</a> of inverse edges or else by inserting or deleting a defining path for one of the 2-cells of the complex. There is an equivalence relation, ∼, on paths in Γ which is generated by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M36">View MathML</a> for any edges <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M37">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M38">View MathML</a> of Γ. This corresponds to requiring the closed paths <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M39">View MathML</a> at the vertex <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M40">View MathML</a> to be the defining paths for the 2-cells of a 2-complex having Γ as its 1-skeleton. This 2-complex is called the Squier complex of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M5">View MathML</a> and denoted by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M42">View MathML</a> (see, for example, [6-9]). The paths in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M42">View MathML</a> can be represented by geometric configurations, called monoid pictures. We assume here that the reader is familiar with monoid pictures (see [[6], Section 4], [[7], Section 1] or [[8], Section 2]). Typically, we will use blackboard bold, e.g., <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M44">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M45">View MathML</a>, ℂ, ℙ, as notation for monoid pictures. Atomic monoid pictures are pictures which correspond to paths of length 1. Write <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M46">View MathML</a> for the atomic picture which corresponds to the edge <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M47">View MathML</a> of the Squier complex. Whenever we can concatenate two paths π and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M30">View MathML</a> in Γ to form the path <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M49">View MathML</a>, then we can concatenate the corresponding monoid pictures ℙ and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M50">View MathML</a> to form a monoid picture <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M51">View MathML</a> corresponding to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M49">View MathML</a>. The equivalence of paths in the Squier complex corresponds to an equivalence of monoid pictures. That is, two monoid pictures ℙ and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M50">View MathML</a> are equivalent if there is a finite sequence of monoid pictures <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M54">View MathML</a> where, for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M55">View MathML</a>, the monoid picture <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M56">View MathML</a> is obtained from the picture <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M57">View MathML</a> either by inserting or deleting a subpicture <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M58">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M44">View MathML</a> is an atomic monoid picture, or else by replacing a subpicture <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M60">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M61">View MathML</a> or vice versa, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M44">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M45">View MathML</a> are atomic monoid pictures.

A monoid picture is called a spherical monoid picture when the corresponding path in the Squier complex is a closed path. Suppose Y is a collection of spherical monoid pictures over <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M64">View MathML</a>. Two monoid pictures ℙ and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M50">View MathML</a> are equivalent relative toY if there is a finite sequence of monoid pictures <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M66">View MathML</a> where, for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M67">View MathML</a>, the monoid picture <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M56">View MathML</a> is obtained from the picture <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M57">View MathML</a> either by the insertion, deletion and replacement operations of the previous paragraph or else by inserting or deleting a subpicture of the form <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M70">View MathML</a> or of the form <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M71">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M72">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M73">View MathML</a>. By definition, a set Y of spherical monoid pictures over <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M5">View MathML</a> is a trivializer of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M75">View MathML</a> if every spherical monoid picture is equivalent to an empty picture relative to Y. By [[7], Theorem 5.1], if Y is a trivializer for the Squier complex, then the elements of Y generate the first homology group of the Squier complex. The trivializer is also called a set of generating pictures. Some examples and more details of the trivializer can be found in [7-14].

For any monoid picture ℙ over <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M64">View MathML</a> and for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M77">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M78">View MathML</a> denotes the exponent sum of R in ℙ which is the number of positive discs labeled by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M15">View MathML</a>, minus the number of negative discs labeled by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M16">View MathML</a>. For a non-negative integer n, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M64">View MathML</a> is said to be n-Cockcroft if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M82">View MathML</a> (modn), (where congruence (mod0) is taken to be equality) for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M25">View MathML</a> and for all spherical pictures ℙ over <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M64">View MathML</a>. Then a monoid ℳ is said to be n-Cockcroft if it admits an n-Cockcroft presentation. In fact, to verify the n-Cockcroft property, it is enough to check for pictures <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M85">View MathML</a>, where Y is a trivializer (see [7,8]). The 0-Cockcroft property is usually just called Cockcroft. In general, we take n to be equal to 0 or a prime p. Examples of monoid presentations with Cockcroft and p-Cockcroft properties can be found in [10].

Suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M86">View MathML</a> is a finite presentation for a monoid ℳ. Then the Euler characteristic<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M87">View MathML</a> is defined by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M88">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M89">View MathML</a> is defined by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M90">View MathML</a>. In unpublished work, Pride has shown that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M91">View MathML</a>. With this background, we define the finite monoid presentation <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M64">View MathML</a> to be efficient if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M93">View MathML</a>, and we define the monoid ℳ to be efficient if it has an efficient presentation. Moreover, a presentation <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M94">View MathML</a> for ℳ is called minimal if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M95">View MathML</a> for all presentations <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M64">View MathML</a> of ℳ. There is also interest in finding inefficient finitely presented monoids since if we can find a minimal presentation <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M94">View MathML</a> for a monoid ℳ such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M94">View MathML</a> is not efficient, then we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M99">View MathML</a> for all presentations <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M100">View MathML</a> defining the same monoid ℳ. Thus, there is no efficient presentation for ℳ, that is, ℳ is not an efficient monoid.

The following theorem was first given in [10]. (The group version of this result was proved by Epstein in [15].)

Theorem 1.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M64">View MathML</a>be a monoid presentation. Then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M64">View MathML</a>is efficient if and only if it isp-Cockcroft for some primep.

Let ℳ be a monoid with the presentation <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M103">View MathML</a>, and let

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M104">View MathML</a>

be the free left <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M105">View MathML</a>-module with basis <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M106">View MathML</a>. For an atomic picture <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M107">View MathML</a> (where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M108">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M109">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M26">View MathML</a>), we define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M111">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M112">View MathML</a>. For any spherical monoid picture ℙ, we then define

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M113">View MathML</a>

(1)

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M114">View MathML</a> be the coefficient of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M115">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M116">View MathML</a>. So, we can write

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M117">View MathML</a>

(2)

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M118">View MathML</a> be a two-sided ideal of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M105">View MathML</a> generated by the elements <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M114">View MathML</a>, where ℙ is a spherical monoid picture and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M109">View MathML</a>. Then this ideal is called the second Fox ideal of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M5">View MathML</a>. More specifically, for a trivializer Y of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M75">View MathML</a>, the set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M118">View MathML</a> is generated (as two-sided ideal) by the elements <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M114">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M126">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M127">View MathML</a>. We note that all this above material given by the consideration ‘left’ can also be applied to ‘right’ for a monoid ℳ.

The definition and a standard presentation for the semi-direct product of two monoids can be found in [10,11,14,16]. Let A and K be arbitrary monoids with associated presentations <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M128">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M129">View MathML</a>, respectively. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M130">View MathML</a> be the corresponding semi-direct product of these two monoids, where θ is a monoid homomorphism from A to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M131">View MathML</a>. (We note that the reader can find some examples of monoid endomorphisms in [17].) The elements of ℳ can be regarded as ordered pairs <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M132">View MathML</a> where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M133">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M134">View MathML</a> with multiplication given by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M135">View MathML</a>. The monoids A and K are identified with the submonoids of ℳ having elements <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M136">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M137">View MathML</a>, respectively. We want to define standard presentations for ℳ. For every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M138">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M139">View MathML</a>, choose a word, which we denote by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M140">View MathML</a>, on Y such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M141">View MathML</a> as an element of K. To establish notation, let us denote the relation <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M142">View MathML</a> on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M143">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M144">View MathML</a> and write t for the set of relations <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M144">View MathML</a>. Then, for any choice of the words <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M140">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M147">View MathML</a>

(3)

is a standard monoid presentation for the semi-direct product ℳ.

In [14], a finite trivializer set has been constructed for the standard presentation <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M148">View MathML</a>, as given in (3), for the semi-direct product ℳ. We will essentially follow [10] in describing this trivializer set using spherical pictures and certain non-spherical subpictures of these.

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M149">View MathML</a> is a positive word on Y, then for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M150">View MathML</a>, we denote the word <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M151">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M152">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M153">View MathML</a> is a positive word on X, then for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M139">View MathML</a>, we denote the word <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M155">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M156">View MathML</a>, and this can be represented by a monoid picture, say <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M157">View MathML</a>, as in Figure 2(b). For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M139">View MathML</a> and the relation <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M159">View MathML</a> in the relation set r, we have two important special cases, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M160">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M161">View MathML</a>, of this consideration. We should note that these non-spherical pictures consist of only <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M144">View MathML</a>-discs (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M150">View MathML</a>). Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M164">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M150">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M166">View MathML</a>, there is a non-spherical picture, say <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M167">View MathML</a>, over <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M168">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M169">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M170">View MathML</a>. Further, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M159">View MathML</a> be a relation <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M25">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M139">View MathML</a>. Since θ is a homomorphism, by the definition on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M156">View MathML</a>, we have that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M175">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M176">View MathML</a> must represent the same element of the monoid K. That is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M177">View MathML</a>. Hence, there is a non-spherical picture over <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M168">View MathML</a> which we denote by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M179">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M180">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M181">View MathML</a>. In fact, there may be many different ways to construct the pictures <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M167">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M179">View MathML</a>. These pictures must exist, but they are not unique. On the other hand, the picture <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M157">View MathML</a> will depend upon our choices for words <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M140">View MathML</a>, but this is unique once these choices are made.

After all, for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M150">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M139">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M25">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M189">View MathML</a>, one can construct spherical monoid pictures, say <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M190">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M191">View MathML</a>, by using the non-spherical pictures <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M167">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M160">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M161">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M179">View MathML</a> (see Figures 2, 3 and 4 for the examples of these pictures). Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M196">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M197">View MathML</a> be trivializer sets of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M198">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M199">View MathML</a>, respectively. Also, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M200">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M201">View MathML</a>. Then, by [10,14], it is known that for a presentation <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M148">View MathML</a>, as in (3), a trivializer set of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M203">View MathML</a> is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M204">View MathML</a>.

2 Generators over the semi-direct product of finite cyclic monoids

In fact, this is the main section of the paper and it will be given as two subsections under the names of Part I and Part II. Since we will define generating functions by considering the exponent sums of the generating pictures over the presentation of this semi-direct product, the first subsection is aimed to define these generating pictures and the related results about them.

2.1 Part I: generating pictures

In this subsection, we will mainly present the efficiency (equivalently, p-Cockcroft property for a prime p by Theorem 1.1) for the semi-direct products of finite cyclic monoids.

Let A and K be two finite cyclic monoids with presentations

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M205">View MathML</a>

(4)

respectively, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M206">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M207">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M208">View MathML</a>, or equivalently,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M209">View MathML</a>

(5)

Due to [10], a trivializer set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M210">View MathML</a> (and similarly <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M196">View MathML</a>) of the Squier complex <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M199">View MathML</a> (and similarly <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M198">View MathML</a>) is given by the pictures <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M214">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M215">View MathML</a>), as in Figure 1.

thumbnailFigure 1. Generating pictures of finite monogenic monoids.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M216">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M217">View MathML</a>) be an endomorphism of K. Then we have a mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M218">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M219">View MathML</a>. In fact this induces a homomorphism <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M220">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M221">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M222">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M223">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M224">View MathML</a> are equal if and only if they agree on the generator y of K, we must have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M225">View MathML</a>

(6)

We then have the semi-direct product <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M226">View MathML</a> and, by [10], a standard presentation

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M227">View MathML</a>

(7)

as in (3), for the monoid M where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M228">View MathML</a>

In the rest of the paper, we will assume that the equality in Equation (6) holds when we talk about the semi-direct product M of K by A.

The subpicture <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M167">View MathML</a> can be drawn as in Figure 2(a), and in fact, by considering this subpicture, we clearly have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M230">View MathML</a>

thumbnailFigure 2. Two subpictures of the generating pictures.

As it is seen in Figure 2(b), we also have the subpicture <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M160">View MathML</a> (and similarly <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M161">View MathML</a>) with

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M233">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M234">View MathML</a>

By equality (6), we must have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M235">View MathML</a>. Hence, by [10], the subpicture <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M179">View MathML</a> with

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M237">View MathML</a>

can be depicted as in Figure 3.

thumbnailFigure 3. Subpicture<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M238">View MathML</a>of the generating picture.

After all, the whole generating pictures <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M190">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M191">View MathML</a> can be drawn as in Figure 4.

thumbnailFigure 4. Collection of the generating pictures of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M241">View MathML</a>in (7).

The following result states necessary and sufficient conditions for the presentation of the split extension of two finite monogenic monoids to be efficient.

Proposition 2.1 ([18])

Letpbe a prime. Suppose that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M242">View MathML</a>is a monoid with the associated monoid presentation<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M243">View MathML</a>, as in (7). Then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M243">View MathML</a>isp-Cockcroft (equivalently efficient) if and only if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M245">View MathML</a>

Remark 2.2 To be an example of Proposition 2.1, one can take

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M246">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M247">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M248">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M249">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M250">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M251">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M252">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M253">View MathML</a> while <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M254">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M255">View MathML</a> to get 2-Cockcroft property for the presentation <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M243">View MathML</a> in (7), or more generally

• for any prime p, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M257">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M258">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M259">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M260">View MathML</a> to get p-Cockcroft property for the presentation <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M243">View MathML</a> in (7).

Considering Theorem 1.1, one can say that the monoid presentation <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M243">View MathML</a>, as in (7), is efficient if and only if there is a prime p such that

In particular, if we choose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M264">View MathML</a> or 2, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M243">View MathML</a> will be inefficient.

Recall that, by the meaning of finite cyclic monoids, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M266">View MathML</a> cannot be equal to 0. We also note that a similar proof for the following result about minimal but inefficiency of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M243">View MathML</a> can be found in [18].

Proposition 2.3LetMbe the semi-direct product ofKbyA, and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M243">View MathML</a>, as in (7), be the presentation forMwhere<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M269">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M270">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M271">View MathML</a>. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M272">View MathML</a>and the subtraction<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M273">View MathML</a>is not even and not equal to 1, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M243">View MathML</a>is minimal but inefficient.

Remark 2.4 To be an example of Proposition 2.3, we can consider the following:

• For an odd positive integer t, one can take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M275">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M276">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M277">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M278">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M272">View MathML</a> in the presentation given in (7). Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M280">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M281">View MathML</a>), the presentation is minimal but inefficient.

• For all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M282">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M283">View MathML</a>, one can take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M284">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M285">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M286">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M260">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M272">View MathML</a> in the presentation given in (7). Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M289">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M281">View MathML</a>), the presentation is minimal while it is inefficient.

2.2 Part II: generating functions

By considering the pictures defined in the previous section and also the evaluations obtained from them, we will define the related generating functions. In another words, by taking into account Propositions 2.1 and 2.3, we will reach our main aim over monoids of this paper.

We firstly recall that, as noted in [[4], Remark 1.1], if a monoid presentation satisfies efficiency or inefficiency (while it is minimal), then it always has a minimal number of generators. Working with the minimal number of elements gives a great opportunity to define related generating functions over this presentation. This will be one of the key points in our results.

Our first result of this section is related to the connection of the monoid presentation in (7) with array polynomials. In fact array polynomials<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M291">View MathML</a> are defined by means of the generating function

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M292">View MathML</a>

(cf.[19-21]). According to the same references, array polynomials can also be defined as the form

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M293">View MathML</a>

(8)

Since the coefficients of array polynomials are integers, they find very large application area, especially in system control (cf.[22]). In fact, these integer coefficients give us the opportunity to use these polynomials in our case. We should note that there also exist some other polynomials, namely Dickson, Bell, Abel, Mittag-Leffler etc., which have integer coefficients which will not be handled in this paper.

From (5), we know that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M294">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M295">View MathML</a>. Hence, by considering the meaning and conditions of Proposition 2.1, we obtain the following theorem as one of the main results of this paper.

Theorem 2.5The efficient presentation<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M11">View MathML</a>defined in (7) has a set of generating functions

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M297">View MathML</a>

(9)

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M291">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M299">View MathML</a>are defined as in (8).

Proof Let us consider the generating pictures <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M190">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M191">View MathML</a> (in Figure 4) with their non-spherical subpictures defined in Figures 2 and 3, and the generating pictures of finite monogenic monoids defined in Figure 1. Recall that by counting the exponent sums of the discs R, S and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M144">View MathML</a> in the related pictures, the conditions of Proposition 2.1 have been obtained [18]. (For more similar results and applications, one can see the papers [10,11].)

To reach our aim in the proof, we first need to calculate <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M303">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M304">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M305">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M215">View MathML</a>) and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M307">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M308">View MathML</a>). By Equations (1) and (2), we have

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M310">View MathML</a> denotes the Fox derivation [23]. Also, for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M215">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M308">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M313">View MathML</a>

Therefore, by the definition, the second Fox ideal <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M314">View MathML</a> of the presentation <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M11">View MathML</a> in (7) is generated by the polynomial elements

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M316">View MathML</a>

(10)

We need to keep our calculations going to other evaluations in the above polynomial elements. To do that, one can consider the augmentation map <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M317">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M318">View MathML</a>. Under this map, it is easy to see that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M319">View MathML</a>

(11)

and for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M215">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M321">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M322">View MathML</a>

Now, by using (8) and keeping in our mind the coefficients of array polynomials are integer, we clearly have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M323">View MathML</a>

Then, by reformulating the elements in (10) and (11) of the second Fox ideal <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M314">View MathML</a>, we arrive at the functions in (9) as desired. □

Considering Remark 2.2, we obtain the following corollary as a consequence of Theorem 2.5.

Corollary 2.6For any primep, the presentation

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M325">View MathML</a>

has a set of generating functions

In Proposition 2.3, the minimality (while satisfying inefficiency) of the presentation <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M11">View MathML</a> was expressed in (7). Thus, by considering the meaning and conditions of Proposition 2.3, we obtain the following theorem as another main result of this paper. Since the proof is quite similar to the proof of Theorem 2.5, we omit it.

Theorem 2.7The inefficient but minimal presentation<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M11">View MathML</a>defined in (7) has a set of generating functions

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M273">View MathML</a>is an odd integer and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M291">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M299">View MathML</a>are defined as in (8).

By considering Remark 2.4, we can have the following consequences of Theorem 2.7.

Corollary 2.8For an odd positive integert, the presentation

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M333">View MathML</a>

has a set of generating functions

Corollary 2.9For any positive integerssandtwith the condition<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M283">View MathML</a>, the presentation

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M336">View MathML</a>

has a set of generating functions

Remark 2.10 Since both presentations in Propositions 2.1 and 2.3 have the minimal number of generators because of their efficiency or inefficiency (but minimal) status, this situation affected very positively the number and type of generating functions defined on them.

At this point, we should note that for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M338">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M339">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M340">View MathML</a>, generalized array type polynomials<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M341">View MathML</a> related to the non-negative real parameters have been recently developed (in [20]) and some elementary properties including recurrence relations of these polynomials have been derived. In fact, by setting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M342">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M343">View MathML</a>, Equation (8) is obtained.

Remark 2.11 For a future project, one can study the generalization of Theorems 2.5 and 2.7 by using <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M344">View MathML</a>.

The remaining goal of this section is to make a connection between the presentation <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M11">View MathML</a> in (7) and Stirling numbers of the second kind (cf.[20,24-28] and the references of these papers). In fact, Stirling numbers of the second kind <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M346">View MathML</a> are defined by means of the generating function

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M347">View MathML</a>

(see [27,28]). According to [[20], Theorem 1, Remark 2], Stirling numbers can also be defined as the form

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M348">View MathML</a>

We remind that these numbers satisfy the well-known properties

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M349">View MathML</a>

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M350">View MathML</a> denotes the Kronecker symbol (see [27,28]). It is known that Stirling numbers are used in combinatorics, in number theory, in discrete probability distributions for finding higher-order moments, etc. We finally note that since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M346">View MathML</a> is the number of ways to partition a set of n objects into k groups, these numbers find an application area in combinatorics and in theory of partitions.

In addition to the above formulas, for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M346">View MathML</a>, by [20,26,27], we also have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M353">View MathML</a>

(12)

as a formula for Stirling numbers. Therefore, by taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M354">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M355">View MathML</a> in Equation (12), the polynomial elements of the second Fox ideal <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M314">View MathML</a> of the presentation <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M11">View MathML</a> in (7) can be restated as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M358">View MathML</a>

(13)

After that, as a different version of Theorem 2.5 (and so Theorem 2.7), we present the following result.

Theorem 2.12The efficient presentation<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M11">View MathML</a>in (7) has a set of generating functions in terms of Stirling numbers as given in (13). By taking<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M272">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M273">View MathML</a>is an odd positive integer, we get a set of generating functions in terms of Stirling numbers for the inefficient but minimal presentation of the form as defined in (7).

Furthermore, in a recent work, Simsek [20] has constructed the generalizedγ-Stirling numbers of the second kind<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M362">View MathML</a> related to non-negative real parameters (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M363">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M364">View MathML</a>, a complex number γ and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M365">View MathML</a>). In fact, this new generalization is defined by the generating function as the equality

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M366">View MathML</a>

(14)

By setting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M367">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M368">View MathML</a> in (14), one can obtain the γ-Stirling numbers of the second kind <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M369">View MathML</a> which are defined by the generating function

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M370">View MathML</a>

(see [27,28]). According to the same references, by substituting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M371">View MathML</a> into the above equation, the Stirling numbers of the second kind <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M372">View MathML</a> are obtained.

By considering this new generalization <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M362">View MathML</a>, in [[20], Theorem 1], the equality

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M374">View MathML</a>

(15)

has also been obtained for γ-Stirling numbers of the second kind. In fact, by setting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M367">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M368">View MathML</a> in (15), one can get the following equality on γ-Stirling numbers:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M377">View MathML</a>

(16)

(see [27,28]).

Hence, we can present the following note.

Remark 2.13 It is clearly seen that Stirling numbers have been only considered in Theorems 2.5 and 2.7 (and the corollaries about them). However, one can also study the γ-Stirling numbers <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M378">View MathML</a> defined in (16) and generalized γ-Stirling numbers <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M362">View MathML</a> defined in (15) to obtain different types of generating functions.

3 The constant function related to main results

In Theorems 2.5, 2.7 and 2.12, we have actually used

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M380">View MathML</a>

as the constants of defined generating functions. In this section, by representing these constants as a single function (see Equation (17) below), we investigate some new properties over it.

Hence, let us consider the analytic function

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M381">View MathML</a>

(17)

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M382">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M383">View MathML</a>. To reach our aim, let us first replace the complex element z by a positive integer i in (17), and then apply some fundamental algebraic progress to it. Therefore,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M384">View MathML</a>

(18)

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M385">View MathML</a> is a cyclotomic polynomial having degree <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M386">View MathML</a>. By considering finite powers of the function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M387">View MathML</a> given in (17), we can get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M388">View MathML</a>

(19)

which is actually m-times algebraic multiplication of the function f.

Now, if we replace z by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M389">View MathML</a>, then we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M390">View MathML</a>

(20)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M391">View MathML</a>

(21)

Further, by applying the Cauchy multiplication in (21), we finally obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M392">View MathML</a>

All these above processes imply the following result.

Theorem 3.1

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M393">View MathML</a>

(22)

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M394">View MathML</a>

and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M395">View MathML</a>denotes the Stirling numbers of second kind.

Some properties of the function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M396">View MathML</a> in (19) can be expressed as follows:

• If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M382">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M398">View MathML</a> is an analytic function, and then it has a power series as defined in the above theorem with Equation (22).

• If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M399">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M398">View MathML</a> is a continuous function which is actually a polynomial function having degree <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M401">View MathML</a>.

• If we replace z by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M389">View MathML</a> in (19), we obtain the second kind Stirling numbers.

Remark 3.2 Setting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M403">View MathML</a> in (22), one can easily see that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M404">View MathML</a>

since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M405">View MathML</a>.

By considering [[20], Eq. (3.2)] and Equation (20), we can extend Remark 3.2 to a general natural number <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M406">View MathML</a> as in the following theorem.

Theorem 3.3

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M407">View MathML</a>

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M408">View MathML</a>denotes the array polynomials.

As it was seen, only the function defined in (18) itself is enough to represent almost all the conditions in Propositions 2.1 and 2.3. Thus, we can express the following remark which depicts some new studying areas for a future project.

Remark 3.4

• If we replace z by i, then we can study the changes on the generating pictures defined in Figures 1, 2, 3 and 4. By playing on this function, one can hope to apply some operations (as defined in [7,8]) on the pictures, and so it could happen to represent these algebraic operations by generating functions to obtain efficiency or inefficiency (while minimality holds).

• While <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M382">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M399">View MathML</a>, analytic and functional equations can be studied.

• As we have partially done in the above, replacing z by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M389">View MathML</a>, one can study the generating functions of array polynomials and Stirling numbers.

3.1 Some other properties over this constant

Let us consider the first derivation of the function in (17). We then have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M412">View MathML</a>

or equivalently,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M413">View MathML</a>

(23)

In (23), replacing z by i, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M414">View MathML</a>

and then by using (18), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M415">View MathML</a>

(24)

As the next step, let us calculate the second derivative of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M416">View MathML</a>:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M417">View MathML</a>

and by collecting some terms in brackets, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M418">View MathML</a>

Now, using (24), the second derivative of the function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M416">View MathML</a> will be equal to

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M420">View MathML</a>

(25)

Replacing z by i in (25) and using (18), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M421">View MathML</a>

By iterating these above derivations for the variable z, and then replacing z by i, we finally obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M422">View MathML</a>

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M423">View MathML</a> stands for some constants.

This above theory is related to the functional equations. In fact, these above progresses show that the presentation <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M11">View MathML</a> in (7) can be related to the functional equations.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors completed the paper together. All authors read and approved the final manuscript.

Acknowledgements

All authors are partially supported by Research Project Offices of Uludağ, Selçuk and Akdeniz Universities, and TUBITAK (The Scientific and Technological Research Council of Turkey).

References

  1. Stillwell, J: Classical Topology and Combinatorial Group Theory, Springer, Berlin (1980)

  2. Birkhoff, GD, Lewis, D: Chromatic polynomials. Trans. Am. Math. Soc.. 60, 355–451 (1946)

  3. Cardoso, DM, Silva, ME, Szymanski, J: A generalization of chromatic polynomial of a graph subdivision. J. Math. Sci.. 183(2), 246–254 (2012)

  4. Cangul, IN, Cevik, AS, Simsek, Y: A new approach to connect algebra with analysis: relationships and applications between presentations and generating functions. Bound. Value Probl. (accepted)

  5. Howie, JM: Fundamentals of Semigroup Theory, Clarendon, New York (1995)

  6. Guba, V, Sapir, M: Diagram Groups, Am. Math. Soc., Providence (1997)

  7. Pride, SJ: Geometric methods in combinatorial semigroup theory. In: Fountain J (ed.) Semigroups, Formal Languages and Groups, pp. 215–232. Kluwer Academic, Dordrecht (1995)

  8. Pride, SJ: Low-dimensional homotopy theory for monoids. Int. J. Algebra Comput.. 5(6), 631–649 (1995). Publisher Full Text OpenURL

  9. Squier, CC: Word problems and a homological finiteness condition for monoids. J. Pure Appl. Algebra. 49, 201–216 (1987). Publisher Full Text OpenURL

  10. Cevik, AS: The p-Cockcroft property of the semidirect products of monoids. Int. J. Algebra Comput.. 13(1), 1–16 (2003). Publisher Full Text OpenURL

  11. Cevik, AS: Minimal but inefficient presentations of the semidirect products of some monoids. Semigroup Forum. 66, 1–17 (2003)

  12. Cremanns, R, Otto, F: Finite derivation type implies the homological finiteness condition <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/15/mathml/M425">View MathML</a>. J. Symb. Comput.. 18, 91–112 (1994). Publisher Full Text OpenURL

  13. Ivanov, SV: Relation modules and relation bimodules of groups, semigroups and associative algebras. Int. J. Algebra Comput.. 1, 89–114 (1991). Publisher Full Text OpenURL

  14. Wang, J: Finite derivation type for semi-direct products of monoids. Theor. Comput. Sci.. 191(1-2), 219–228 (1998). Publisher Full Text OpenURL

  15. Epstein, DBA: Finite presentations of groups and 3-manifolds. Q. J. Math.. 12, 205–212 (1961). Publisher Full Text OpenURL

  16. Saito, T: Orthodox semi-direct products and wreath products of monoids. Semigroup Forum. 38, 347–354 (1989). Publisher Full Text OpenURL

  17. Dlab, V, Neumann, BH: Semigroups with few endomorphisms. J. Aust. Math. Soc. A. 10, 162–168 (1969). Publisher Full Text OpenURL

  18. Ates, F, Cevik, AS: Minimal but inefficient presentations for semi-direct products of finite cyclic monoids. Groups St. Andrews 2005, Volume 1. 170–185 (2007)

  19. Chang, CH, Ha, CW: A multiplication theorem for the Lerch zeta function and explicit representations of the Bernoulli and Euler polynomials. J. Math. Anal. Appl.. 315, 758–767 (2006). Publisher Full Text OpenURL

  20. Simsek, Y: Generating functions for generalized Stirling type numbers, array type polynomials, Eulerian type polynomials and their applications. arXiv:1111.3848v2 [math.NT] 23 Nov 2011

  21. Simsek, Y: Interpolation function of generalized q-Bernstein type polynomials and their application. Curves and Surfaces, pp. 647–662. Springer, Berlin (2011)

  22. Mismar, MJ, Abu-Al-Nadi, DI, Ismail, TH: Pattern synthesis with phase-only control using array polynomial technique. Dubai, UAE, 24-27 November 2007. (2007)

  23. Brown, KS: Cohomology of Groups, Springer, Berlin (1982)

  24. Agoh, T, Dilcher, K: Shortened recurrence relations for Bernoulli numbers. Discrete Math.. 309, 887–898 (2009). Publisher Full Text OpenURL

  25. Carlitz, L: Some numbers related to the Stirling numbers of the first and second kind. Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. Fiz.. 544-576, 49–55 (1976)

  26. Kim, T: q-Volkenborn integration. Russ. J. Math. Phys.. 19, 288–299 (2002)

  27. Luo, QM, Srivastava, HM: Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind. Appl. Math. Comput.. 217, 5702–5728 (2011). Publisher Full Text OpenURL

  28. Srivastava, HM: Some generalizations and basic (or q-) extensions of the Bernoulli, Euler and Genocchi polynomials. Appl. Math. Inf. Sci.. 5, 390–444 (2011)