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

Open Access Research

A new approach to connect algebra with analysis: relationships and applications between presentations and generating functions

Ismail Naci Cangül1*, Ahmet Sinan Çevik2 and Yılmaz Şimşek3

Author affiliations

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

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

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

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Citation and License

Boundary Value Problems 2013, 2013:51  doi:10.1186/1687-2770-2013-51

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/51


Received:1 December 2012
Accepted:14 February 2013
Published:14 March 2013

© 2013 Cangül 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

For a minimal group (or monoid) presentation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1">View MathML</a>, let us suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1">View MathML</a> satisfies the algebraic property of either being efficient or inefficient. Then one can investigate whether some generating functions can be applied to it and study what kind of new properties can be obtained by considering special generating functions. To establish that, we will use the presentations of infinite group and monoid examples, namely the split extensions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M4">View MathML</a>, respectively. This study will give an opportunity to make a new classification of infinite groups and monoids by using generating functions.

MSC: 11B68, 11S40, 12D10, 20M05, 20M50, 26C05, 26C10.

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

1 Introduction and preliminaries

In the literature, although there are so many studies about figuring out the relationship between rings (or fields) and special generating functions (cf., for instance, [1-4]), there are no such studies about the relationship between group (or monoid) presentations and generating functions. In fact, the studies on the efficient and inefficient (but minimal) group and monoid presentations gave very important characterisations for groups and monoids in the branch of combinatorial group theory of mathematics (see, for instance, [5-12]). It is known that generating functions are still interesting for many mathematicians and physicians (see, for instance, [2,13,14] in addition to above). Thus, it would be quite interesting for future studies to connect these two important areas and then search for possible properties.

In the light of this thought, in this paper, a connection between special (efficient and inefficient) presentations defined on infinite groups (and monoids) and some generating functions related to the special polynomials and numbers will be investigated. (These special polynomials are chosen by their integer coefficients. Of course, one can choose some other polynomials used in this paper.) Another aim of this paper is to try to make a classification of infinite groups and monoids.

This paper is divided into four sections. Main results are presented specially in Sections 2 and 3. In the remaining parts of this section, we will present some fundamental material related to the group or monoid presentations that will be needed in later sections of this paper.

A group (or a monoid) presentation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M5">View MathML</a>

(1)

is a pair where x is a set (generating symbols) and r is a set of non-empty, cyclically reduced words (relators) on x. In monoids, each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M6">View MathML</a> is actually an ordered pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M7">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M9">View MathML</a> are distinct, positive (one of them could be empty) words on x. We say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1">View MathML</a> is finite if x and r are both finite. Further, all results in this paper are related to split extensions and their presentations. In [15], a split extension is also named a semidirect product and detailed properties of this product can be found in elementary algebra textbooks. Here, we will just remind the presentation of a semidirect product of arbitrary groups (or monoids). Therefore, for arbitrary groups (or monoids) A and K with presentations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M12">View MathML</a>, the presentation of the group (or monoid) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M13">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M14">View MathML</a>

(2)

where t is the set of relators of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M15">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M16">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M17">View MathML</a> (cf.[8,9]). We remind that the homomorphism θ is defined from A to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M18">View MathML</a> for the semidirect product of groups, while it is defined from A to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M19">View MathML</a> for the product of monoids. Further, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M20">View MathML</a> is an isomorphism of the group K and a homomorphism in a monoid case.

In the next two subsections, we will give some other preliminary material that will be needed for the construction of the results in this paper by considering the presentation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1">View MathML</a> in (1).

1.1 Efficiency

The subject under this title will be given over a group G with a presentation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1">View MathML</a> as defined in (1). But we should note that the following material will be completely the same if the group G is replaced by a monoid M.

For the presentation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1">View MathML</a>, the Euler characteristic is defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M24">View MathML</a>. By [16-18], there exists a lower bound <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M25">View MathML</a> which is equal to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M26">View MathML</a> with the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M27">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M28">View MathML</a> denotes the ℤ-rank of the torsion-free part and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M29">View MathML</a> denotes the minimal number of generators. Depending on these numbers, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M30">View MathML</a>

Therefore a presentation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1">View MathML</a> is called minimal if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M32">View MathML</a> for all presentations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M33">View MathML</a> of G, or is called efficient if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M34">View MathML</a>. Moreover, G is called efficient if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M35">View MathML</a>. In [7,8], Cevik recalled known results for efficiency of groups and monoids. (We should remark that some authors also consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M36">View MathML</a> and call this the deficiency of the presentation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1">View MathML</a>.)

Remark 1 In both group and monoid cases, if the presentation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1">View MathML</a> in (1) is efficient or inefficient while it is minimal, then it always has a minimal number of generators. So, this fact affects positively the use of generating functions for this type of presentations since we have a great advantage to work with quite a limited number of variables in such a generating function.

1.2 Pictures

There exists a geometric method called spherical group (or monoid) pictures related to the presentation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1">View MathML</a> given in (1). This method was constructed and first used by Pride [6,17-20] for both groups and monoids, and since then it has still been in use for the solution of many important combinatorial problems such as word problems (cf.[21,22]). Here, we will recall a brief description of pictures for groups and monoids in separate cases. Before that, we express the following remark.

Remark 2 Similarly to (undirected) graphs, this geometric configuration has a large application area, especially in engineering sciences. For example, the plan of electrical network for a city or the behaviour of DNA molecules in a human body can be figured out with pictures (see Figures 1, 2 and 3).

thumbnailFigure 1. Generating pictures of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M40">View MathML</a>as given in (6).

thumbnailFigure 2. The single generating picture of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M41">View MathML</a>given in (9).

thumbnailFigure 3. The single generating picture of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M41">View MathML</a>in (12).

Pictures for groups: As we depicted in Remark 2, a group picture ℙ over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1">View MathML</a> is a geometric configuration consisting of the following:

• A disc <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M44">View MathML</a> with a basepoint O on the boundary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M45">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M44">View MathML</a>.

• Disjoint discs <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M47">View MathML</a> in the interior of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M44">View MathML</a>. Each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M49">View MathML</a> has a basepoint <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M50">View MathML</a> on the boundary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M51">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M49">View MathML</a>.

• A finite number of disjoint arcs <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M53">View MathML</a>, where each arc lies in the closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M54">View MathML</a> and is either a simple closed curve having trivial intersection with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M55">View MathML</a>, or is a simple non-closed curve which joins two points of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M56">View MathML</a>, neither point being a basepoint. Each arc has a normal orientation indicated by a short arrow meeting with the arc transversely and is labelled by an element of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M57">View MathML</a> which is called the label of the arc.

• If we travel around <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M51">View MathML</a> once in the clockwise direction starting from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M50">View MathML</a> and read off the labels on arcs encountered (if we cross an arc, labelled x say, in the direction of its normal orientation, then we read x, whereas if we cross the arc in the direction of its opposite orientation, then we read <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M60">View MathML</a>), then we obtain a word which belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M61">View MathML</a>. We call this word the label of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M49">View MathML</a>. If s is a subset of r, then a disc labelled by an element of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M63">View MathML</a> is called an s-disc.

When we refer to the discs of ℙ, we in fact mean the discs <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M64">View MathML</a>, and not the ambient disc <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M44">View MathML</a>. A closed arc which encircles neither a disc nor an arc of ℙ is called a floating circle. We define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M66">View MathML</a> to be <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M45">View MathML</a>. The label on ℙ (denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M68">View MathML</a>) is the word read off by travelling around <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M66">View MathML</a> once in the clockwise direction starting from O. (In fact, this fact on pictures implies the fundamentals of solving the word problem [21,22].)

Further, ℙ is called spherical if no arcs meet <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M70">View MathML</a> (i.e. if ℙ is spherical, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M70">View MathML</a> is omitted). A transverse pathγ in a picture ℙ is a path in the closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M54">View MathML</a> which intersects the arcs of ℙ only finitely many times. Reading off the labels on the arcs encountered while travelling along a transverse path from its initial point to its terminal point gives a word on x denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M73">View MathML</a>. Let γ be a simple closed transverse path in ℙ. The part of ℙ enclosed by γ is called a subpicture of ℙ. If γ intersects no arcs, then the part of ℙ enclosed by γ is called a spherical subpicture of ℙ. A cancelling pair in ℙ is a spherical subpicture with exactly two discs whose basepoints lie in the same region.

A spray for ℙ is a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M74">View MathML</a> of simple transverse paths satisfying the following: for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M75">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M76">View MathML</a> starts at O and ends at the basepoint of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M49">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M76">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M80">View MathML</a> intersect only at O; travelling around O clockwise in ℙ, we encounter these transverse paths <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M81">View MathML</a>, respectively.

There are some elementary operations (deletion and insertion of a floating circle, deletion and insertion of a cancelling pair, bridge move) on spherical pictures. Then two spherical pictures are called equivalent if one can be obtained from the other by a finite number of above operations. These operations imply an equivalence relation and the equivalence class containing ℙ which is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M82">View MathML</a>. The set of all equivalence classes of spherical pictures over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M83">View MathML</a> forms an abelian group. In addition, for a word W on x, a new spherical picture over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1">View MathML</a> denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M85">View MathML</a> can also be obtained from W by surrounding ℙ with a collection of concentric arcs with total label W. Hence, there is a well-defined <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M86">View MathML</a>-action on equivalence classes of spherical pictures given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M87">View MathML</a> (where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M88">View MathML</a>). We then obtain a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M89">View MathML</a>-module <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M90">View MathML</a> called the second homotopy module of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1">View MathML</a>. Let X be a set of spherical pictures. Then we say that X generates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M92">View MathML</a> (or X is a set of generating pictures) if the elements <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M82">View MathML</a> (where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M94">View MathML</a>) generate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M92">View MathML</a>.

For any picture ℙ over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1">View MathML</a> and for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M97">View MathML</a>, the exponent sum of R in ℙ, denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M98">View MathML</a>, is the number of discs of ℙ labelled by R minus the number of discs labelled by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M99">View MathML</a>. We remind that if pictures <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M100">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M101">View MathML</a> are equivalent, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M102">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M103">View MathML</a>. Depending on the exponent sum, we have the following definition.

Definition 1 For a non-negative integer n, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1">View MathML</a> is said to be n-Cockcroft if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M105">View MathML</a> (where congruence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M106">View MathML</a> is taken to be equality) for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M107">View MathML</a> and for all spherical pictures ℙ over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1">View MathML</a>. Moreover, a group G is said to be n-Cockcroft if it admits an n-Cockcroft presentation.

Actually, to verify that the n-Cockcroft property holds, it is enough to check it only for pictures <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M94">View MathML</a>, where X is a set of generating pictures. Also, the 0-Cockcroft property is usually just called Cockcroft, and in practice, n is taken as a prime p or 0. By [16,19], the presentation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1">View MathML</a> is efficient if and only if it is p-Cockcroft for some prime p. So, this connection between efficiency and p-Cockcroft property will be one of the main ideas during the construction of this paper.

There is an embedding μ of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M92">View MathML</a> into the free module <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M112">View MathML</a> defined as follows (see [6,19]): Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M113">View MathML</a> and suppose that ℙ has discs <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M114">View MathML</a> with the labels <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M115">View MathML</a>, respectively (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M116">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M117">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M75">View MathML</a>). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M119">View MathML</a> be a spray defined previously. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M120">View MathML</a>

(3)

For simplicity, the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M121">View MathML</a> will be preferred instead of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M122">View MathML</a>. For each spherical picture ℙ over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1">View MathML</a> and for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M97">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M125">View MathML</a> be the coefficients of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M126">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M121">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M128">View MathML</a> be the two-sided ideal in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M129">View MathML</a> generated by the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M130">View MathML</a>. This ideal is called the second Fox ideal of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1">View MathML</a>. The concept of Fox ideals has been discussed in [23]. In fact, we need this concept for our studies in this paper as our main goal in this paper is to establish a relationship between generating functions and presentations. For the group case in Section 2, the generating functions will be labelled by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M132">View MathML</a> defined in (3).

Pictures for monoids. As we pointed out in the beginning of this section, some of the following material may also be found in [11,17,18,20]. For a monoid M, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1">View MathML</a> be a monoid presentation as in (1); and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M134">View MathML</a> be a free monoid on x. If we have an element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M135">View MathML</a> (where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M136">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M137">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M138">View MathML</a>) of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M134">View MathML</a>, then we can replace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M140">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M141">View MathML</a> to get a word <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M142">View MathML</a>. This can be represented by a geometric object called an atomic (monoid) picture<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M143">View MathML</a> as depicted in Figure 4.

thumbnailFigure 4. An atomic monoid picture.

We remark that the disc labelled by S in an atomic picture <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M144">View MathML</a> is said to be positive if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M145">View MathML</a> and is said to be negative if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M146">View MathML</a>.

We have a graph Γ (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M147">View MathML</a>) associated with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1">View MathML</a>, called the Squier graph, which is defined as follows: The vertex set is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M134">View MathML</a>, and the edge set is the collection of all atomic monoid pictures. For an orientation of Γ, we will take all edges <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M150">View MathML</a>. For an atomic picture <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M144">View MathML</a>, as in Figure 4, the word we read off by travelling along the top of the atomic picture from left to right gives the initial function, denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M152">View MathML</a>, and the word we read off by travelling along the bottom gives the terminal function, denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M153">View MathML</a>. Also, the mirror image of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M144">View MathML</a> is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M155">View MathML</a>. A path<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M156">View MathML</a> (where each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M157">View MathML</a> is an atomic picture for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M75">View MathML</a>) in Γ will also be called a monoid picture over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M83">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M160">View MathML</a>, then ℙ is called a spherical monoid picture over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1">View MathML</a>. Note that we also have the term subpicture of monoid pictures.

There is a left action of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M134">View MathML</a> on Γ defined as follows. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M163">View MathML</a>.

(i) Let W be a vertex of Γ. Then we define CWto be <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M164">View MathML</a> (product in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M134">View MathML</a>).

(ii) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M144">View MathML</a>, as in Figure 4, be an edge of Γ. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M167">View MathML</a>.

We can define a similar right action of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M134">View MathML</a> on Γ. The left and right actions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M134">View MathML</a> on Γ extend to actions on pictures. That is, if ℙ is a picture and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M170">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M171">View MathML</a>.

For atomic monoid pictures <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M144">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M173">View MathML</a>, one can introduce some operations (deletion and insertion of inverse pairs of atomic pictures and a replacement operation (cf.[17,18])) on spherical monoid pictures. These operations imply an equivalence relation on paths. Therefore the graph Γ with this equivalence relation on paths is called the Squier complex of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1">View MathML</a> denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M175">View MathML</a>. Let Y be a set of spherical monoid pictures. Two spherical monoid pictures will be said to be equivalent (relative toY) if one can be transformed into the other by a finite number of above operations. By [20], the set Y is called a trivializer of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M175">View MathML</a> if every spherical picture is equivalent to an empty picture (relative to Y). Some examples and the details of the trivializer can be found, for instance, in [11,12]. Similarly as in the group case, for any monoid picture ℙ over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1">View MathML</a> and for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M137">View MathML</a>, the exponent sum of S in ℙ is the number of positive discs labelled by S minus the number of negative discs labelled by S. Then the monoid version of Definition 1 can be obtained in completely the same way by replacing the term group with monoid. To verify that the n-Cockcroft (in fact n is taken as a prime p or 0) property holds, it is enough to check it for pictures <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M179">View MathML</a>, where Y is a trivializer of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M175">View MathML</a>.

Let M be a monoid with the presentation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1">View MathML</a> as in (1). Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M182">View MathML</a>

be a free left <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M183">View MathML</a>-module with basis <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M184">View MathML</a>. For an atomic picture <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M185">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M136">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M137">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M138">View MathML</a>, we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M189">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M190">View MathML</a> as in Figure 4. For any spherical monoid picture ℙ, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M191">View MathML</a>

(4)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M192">View MathML</a> be the coefficient of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M193">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M194">View MathML</a>. So, we can write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M195">View MathML</a>

(5)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M196">View MathML</a> be the two-sided ideal of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M183">View MathML</a> generated by the elements <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M192">View MathML</a>, where ℙ is a spherical monoid picture and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M137">View MathML</a>. Then this ideal is called the second Fox ideal of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1">View MathML</a>. More specifically, for a trivializer Y of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M175">View MathML</a>, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M196">View MathML</a> is generated (as two-sided ideal) by the elements <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M192">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M204">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M137">View MathML</a>. We note that all this above material given with the consideration ‘left’ can also be applied to ‘right’ for a monoid M.

In Section 3, the generating functions will be connected to the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M206">View MathML</a> part in (4) or, equivalently, to the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M192">View MathML</a> in (5).

2 The group case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M208">View MathML</a>

Let us consider the split extension <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M209">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M210">View MathML</a> is the a group with rank one, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M211">View MathML</a> is a cyclic group of order n and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M212">View MathML</a> is a homomorphism. Then, by (2), G has the presentation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M213">View MathML</a>

(6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M214">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M215">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M216">View MathML</a>. In [[24], Theorem 3.2.1], the generating set of the second homotopy module <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M217">View MathML</a> has been constructed as drawn in Figure 1. In this generating set, there are two spherical pictures <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M100">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M101">View MathML</a>. In <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M100">View MathML</a>, we have two <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M221">View MathML</a>-discs (one of them is positive and the other is negative), and in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M222">View MathML</a> we have a negative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M221">View MathML</a>-disc and k-times positive <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M221">View MathML</a>-discs. Furthermore, again in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M101">View MathML</a>, there is a total of n-times <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M226">View MathML</a>-discs. Then, by considering the number of discs in these pictures, Baik [[24], Theorem 3.3.3] proved the following result.

Proposition 1The presentation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M227">View MathML</a>in (6) is efficient (equivalently, p-Cockcroft for any primep) if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M228">View MathML</a>.

Therefore, if we suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M229">View MathML</a>, then we obtain an inefficient presentation. Clearly, n must be an odd prime and the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M227">View MathML</a> given in (6) be an inefficient presentation. Otherwise, by setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M231">View MathML</a> in this inefficient case, we obtain the direct product <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M232">View MathML</a> which is a special case of the semidirect products and will not be considered in this paper. By Remark 1, it is always true that efficient presentations (even for groups or monoids) are minimal. But to check the minimality of a presentation while it is inefficient is important, because in this case we obtain the inefficiency of the related group that has this presentation (see [8-10]). For the group case, this important subject is investigated by the following ‘minimality test’ due to Lustig [23].

Lemma 1 ([23])

For any groupGwith a presentation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1">View MathML</a>as in (1), suppose there is a ring homomorphismψfrom<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M129">View MathML</a>into the matrix ring of all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M235">View MathML</a>-matrices (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M236">View MathML</a>) over some commutative ringwith 1. Suppose also that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M237">View MathML</a>. Ifψmaps the second Fox ideal<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M128">View MathML</a>to 0 (in other words, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M128">View MathML</a>is contained in the kernel ofψ), then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1">View MathML</a>is minimal.

By considering Proposition 1, the first main result of this paper is presented as follows.

Theorem 1Let us consider the presentation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M227">View MathML</a>as in (6) for the group<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M242">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M216">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M215">View MathML</a>but<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M245">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M227">View MathML</a>has a set of generating functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M247">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M248">View MathML</a>denotes thenth cyclotomic polynomial overdefined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M249">View MathML</a>

(7)

having a degree<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M250">View MathML</a>.

Proof We first note that since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M227">View MathML</a> is presented as in (6), the action in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M252">View MathML</a> is defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M253">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M254">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M255">View MathML</a>.

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M256">View MathML</a>. Then, by Proposition 1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M227">View MathML</a> is an efficient presentation and so, by Remark 1, is minimal (i.e. has a minimal number of generators). Let us consider the pictures <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M100">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M101">View MathML</a> in Figure 1. Now, by (3), we have

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M254">View MathML</a>, but <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M256">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M216">View MathML</a>. For simplicity, by omitting the overlines on the elements in the above equalities, we obtain that the second Fox ideal is generated by the polynomial elements <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M264">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M265">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M266">View MathML</a>. Now, we can reformulate these polynomial elements as generating functions. It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M267">View MathML</a> has the root <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M268">View MathML</a>. On the other hand, since we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M269">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M270">View MathML</a> has a root t, where t is the multiplicative inverse of k.

Finally, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M271">View MathML</a> has a root <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M268">View MathML</a> modulo n which gives (7) directly. □

Let us take n as an odd prime p. Then, by Proposition 1 and Lemma 1, we get an inefficient but minimal presentation. Thus we have the following corollary.

Corollary 1For an odd primepand a positive integer<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M273">View MathML</a>, the presentation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M227">View MathML</a>in (6) has a set of generating functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M275">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M276">View MathML</a>has a degree of an even number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M277">View MathML</a>.

Remark 3 Theorem 1 and Corollary 1 imply that by choosing the efficient or inefficient minimal presentations, we can get different constants (i.e. the cases of k in both results) and different powers (i.e.n to be a positive integer or an odd prime) in the set of generating functions. Therefore, the structure of the presentation (i.e. efficient or inefficient) affects getting different types of generating functions.

The following consequence of Theorem 1 points out another connection between the presentation in (6) as defined in either Theorem 1 or Corollary 1 and generating functions.

Corollary 2The polynomial<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M278">View MathML</a>in Theorem 1 (or Corollary 1) is actually a ‘locally constant function’.

Proof We recall that the family of locally constant functions [14] is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M279">View MathML</a>

for which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M280">View MathML</a> holds. Moreover, in the meaning of group homomorphisms, each function in this family satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M281">View MathML</a>

Now, by replacing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M282">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M283">View MathML</a>

(8)

it is clear that we get a locally constant function, as required. □

After Theorem 1, Corollary 1 and Corollary 2, we can express the following connection between the generating functions and (twisted) Bernoulli numbers.

Remark 4 The locally constant function corresponding to the generating function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M278">View MathML</a> of the presentation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M227">View MathML</a> given in (6) is related to the twisted Bernoulli numbers and polynomials. (We may refer the reader, for example, to [3,14,25,26] for the twisted Bernoulli numbers and polynomials.) In the next paragraph, we give a brief description.

According to [4,13,14], for each integer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M286">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M287">View MathML</a> denotes the multiplicative group of the primitive <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M288">View MathML</a>th roots of unity in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M289">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M290">View MathML</a>

The dual of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M291">View MathML</a> in the sense of p-adic Pontryagin duality is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M292">View MathML</a>, the direct limit (under inclusion) of cyclic groups <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M287">View MathML</a> of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M288">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M286">View MathML</a>, with discrete topology. The <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M296">View MathML</a> admits a natural <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M291">View MathML</a>-module structure which is written as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M298">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M299">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M300">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M296">View MathML</a> can be embedded discretely in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M302">View MathML</a> as the multiplicative p-torsion subgroup. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M299">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M304">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M305">View MathML</a>, is a locally constant character which is actually a locally analytic character if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M306">View MathML</a>. Then, by [4,13,14,27,28], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M307">View MathML</a> has a continuation to a continuous group homomorphism from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M308">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M309">View MathML</a>. We further remind that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M310">View MathML</a>, then ξ will be assumed to have an rth root of unity with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M311">View MathML</a>.

3 The monoid case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M312">View MathML</a>

Before presenting this special case, let us first discuss a more general situation for the p-Cockcroft property of semidirect products of monoids. In [8,9], by considering a similar version of the picture <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M313">View MathML</a> in Figure 2, the second author investigated the p-Cockcroft property by using the trivializer for the semidirect product <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M314">View MathML</a>, where K and A are arbitrary monoids. (It is seen that there is a single non-spherical subpicture <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M315">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M313">View MathML</a>. In fact, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M315">View MathML</a> contains only S-discs. For an illustration, see Figure 3.) As a special case of it, let us assume that K is a one-relator monoid and A is an infinite cyclic monoid ℤ with presentations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M318">View MathML</a>

respectively. Suppose ψ is an endomorphism of K. Then the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M319">View MathML</a> induces a homomorphism <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M320">View MathML</a>, and we can form the semidirect product <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M321">View MathML</a>. By (2), this product has a presentation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M322">View MathML</a>

(9)

where, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M17">View MathML</a>, the set t is the set of relators

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M324">View MathML</a>

such that the relator S satisfies the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M325">View MathML</a> (or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M326">View MathML</a>). In [8], the necessary and sufficient conditions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M327">View MathML</a> to be efficient are determined.

In the special case above, let us take K as a free abelian monoid of rank two (i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M328">View MathML</a>) presented by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M329">View MathML</a>, and let ψ be the endomorphism <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M330">View MathML</a>, where M is the matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M331">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M332">View MathML</a>) given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M333">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M334">View MathML</a>. As a special case of the presentation in (9), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M335">View MathML</a>

(10)

for the monoid <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M336">View MathML</a> (see [9]). Again, in the same reference, the second author figured out the efficiency of the above presentation as in the following proposition.

Proposition 2 ([9])

For any primep, the presentation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M327">View MathML</a>in (10) isp-Cockcroft if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M338">View MathML</a>.

According to Proposition 2, in particular, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M339">View MathML</a> is not efficient if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M340">View MathML</a> or 2. Therefore the following proposition is proved in the same manner.

Proposition 3 ([9])

The presentation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M327">View MathML</a>in (10) is minimal but inefficient if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M342">View MathML</a>.

The proof of Proposition 3 is based on the following Pride result, which is a monoid version of Lemma 1. Although this result has not been published yet, it has been used in many papers (see, for instance, [8-10]).

Lemma 2 (Pride)

For any monoidMwith a presentation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1">View MathML</a>as in (1), letψbe a ring homomorphism from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M183">View MathML</a>into the ring of all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M345">View MathML</a>-matrices (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M236">View MathML</a>) over some commutative ringwith 1, and suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M237">View MathML</a>. If the second Fox ideal<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M196">View MathML</a>is contained in the kernel ofψ, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M349">View MathML</a>is minimal.

From now on, by considering Propositions 2 and 3, we will reach our main aim of this paper for monoids.

Our first result in this section gives the connection between a monoid presentation and array polynomials. In fact the array polynomials<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M350">View MathML</a> are defined by means of the following generating function:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M351">View MathML</a>

(cf.[29-31]). According to the same references, array polynomials can also be defined in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M352">View MathML</a>

(11)

Since the coefficients of array polynomials are integers, these polynomials find a very large application area, especially in engineering. Array polynomials are used, for instance, in system control (cf.[32]).

In fact these integer coefficients give us an 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. But, since array polynomials have a larger application area in science, we have preferred them. Hence, by considering Proposition 3, we obtain the following theorem as another main result.

Theorem 2Let us consider the monoid<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M353">View MathML</a>with a presentation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M354">View MathML</a>

(12)

Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M327">View MathML</a>has a set of generating functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M356">View MathML</a>

(13)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M350">View MathML</a>is defined as in (11).

Proof Let us consider the spherical picture <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M358">View MathML</a> with its non-spherical subpicture <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M359">View MathML</a> as drawn in Figure 3. In fact, by [9], this is the only picture in the trivializer of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M360">View MathML</a>.

In presentation (12), let us label the relators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M361">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M362">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M363">View MathML</a> by S, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M364">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M365">View MathML</a>, respectively. It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M366">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M367">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M368">View MathML</a>. In the calculation of these exponent sums, we included the exponent sums of S-discs in the non-spherical picture <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M359">View MathML</a>. Actually, a simple calculation shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M370">View MathML</a> and so, by our assumption about <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M327">View MathML</a> that is not efficient, we expect <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M372">View MathML</a> to be 2.

Now, by (4) and (5), the evaluation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M358">View MathML</a> is determined as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M374">View MathML</a>

Therefore, by the definition, the second Fox ideal <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M375">View MathML</a> of the presentation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M327">View MathML</a> in (12) is generated by the polynomial elements

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M377">View MathML</a>

For simplicity, let us replace each of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M378">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M379">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M380">View MathML</a> by 2a, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M381">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M382">View MathML</a>, respectively. In [9], by considering Lemma 2, it has been showed that this presentation in (12) is minimal.

Now, by using (11) and keeping in our mind that the coefficients of array polynomials are integer, we clearly have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M383">View MathML</a>

Then, by reformulating the elements of the second Fox ideal <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M375">View MathML</a>, we arrive at the functions in (13) as desired. □

By considering Proposition 2, if we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M385">View MathML</a>, then we get an efficient presentation. So, for an even prime p, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M386">View MathML</a>. Then one of the presentations of the similar form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M327">View MathML</a> as in (12) can be taken as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M388">View MathML</a>

(14)

which will be efficient. The same procedure in the proof of Theorem 2 gives us the set of generating functions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M327">View MathML</a> in (14) in the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M267">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M391">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M392">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M393">View MathML</a> and the others are defined in (13) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M350">View MathML</a> is given in (11). Nevertheless, by induction steps, we can generalise this last presentation as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M395">View MathML</a>

(15)

Hence we get the following version of Theorem 2 which deals with efficient presentations.

Theorem 3Let us consider the presentation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M327">View MathML</a>in (15) for the monoid<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M397">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M327">View MathML</a>has a set of generating functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M399">View MathML</a>

(16)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M385">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M350">View MathML</a>is defined as in (11).

Remark 5 According to the expression in Remark 1, presentations given in (12), (14) or (15) have a minimal number of generators. But we classified these presentations according to their efficiency status separately in Theorem 2 and Theorem 3. The aim of this separation is to find a solution for a general remark depicted in the final section about obtaining a method for a minimality test by using generating functions (see Section 4 below).

At this point, we should note that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M402">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M403">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M404">View MathML</a>, generalised array type polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M405">View MathML</a> which are related to the non-negative real parameters have been recently developed and some elementary properties including recurrence relations of these polynomials have been obtained [30]. In fact, by setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M406">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M407">View MathML</a>, the equation (11) is obtained.

Remark 6 One can try to study the generalisation of Theorem 2 by using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M408">View MathML</a>.

The remaining goal of this section is to establish a connection between the presentation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M327">View MathML</a> in (12) or (15) and Stirling numbers of the second kind (cf.[3,30,33-36]). In fact, Stirling numbers of the second kind <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M410">View MathML</a> are defined by means of the following generating function:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M411">View MathML</a>

(see [3,36]). According to [[30], Theorem 1, Remark 2], Stirling numbers can also be defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M412">View MathML</a>

We remind that these numbers satisfy the well-known properties

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M413">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M414">View MathML</a> denotes the Kronecker symbol (see [3,36]). 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.boundaryvalueproblems.com/content/2013/1/51/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M410">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 the theory of partitions.

In addition to the above formulas for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M410">View MathML</a>, by [30,35,36], we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M417">View MathML</a>

(17)

as a formula for Stirling numbers. Therefore, in equation (17) by replacing x with a, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M381">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M382">View MathML</a>, respectively, and taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M420">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M421">View MathML</a>, the polynomial elements of the second Fox ideal <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M375">View MathML</a> of the presentation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M327">View MathML</a> in (12) can be restated as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M424">View MathML</a>

(18)

As a different version of Theorem 2, we express the following corollary.

Corollary 3The presentation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M327">View MathML</a>in (12) has a set of generating functions in terms of Stirling numbers as given in (18).

We note that the above corollary can also be stated for the presentation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M327">View MathML</a> in (15).

Furthermore, in a recent work, Simsek [30] has constructed the generalisedλ-Stirling numbers of the second kind<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M427">View MathML</a> related to non-negative real parameters (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M428">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M429">View MathML</a>, λ is a complex number and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M430">View MathML</a>). In fact, this new generalisation is defined by the generating function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M431">View MathML</a>

(19)

By setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M268">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M433">View MathML</a> in (19), one can obtain the λ-Stirling numbers of the second kind <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M434">View MathML</a> which are defined by the generating function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M435">View MathML</a>

(see [3,36]). By substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M436">View MathML</a> into the above equation, the Stirling numbers of the second kind <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M437">View MathML</a>are obtained.

By considering this new generalisation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M427">View MathML</a>, in [[30], Theorem 1], it has been obtained that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M439">View MathML</a>

(20)

for λ-Stirling numbers of the second kind. In fact, by setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M268">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M433">View MathML</a> in (20), one can get the following equality on λ-Stirling numbers:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M442">View MathML</a>

(21)

(see [3,36]).

Hence we can present the following notes about this section:

Remark 7 It is clearly seen that in Theorems 2, 3 and Corollary 3, only Stirling numbers are considered. However, one can also study the λ-Stirling numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M434">View MathML</a> defined in (21) and generalised λ-Stirling numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M427">View MathML</a> defined in (20) as stated in these theorems and corollaries.

Remark 8 For a suitable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M445">View MathML</a> matrix, it is possible to define the presentation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M327">View MathML</a> in (9) (or in (10)) for the monoid <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M447">View MathML</a>. Thus one can try to transform all studies in Section 3 to this general case.

4 Final remarks

In this section we will express some other remarks depicted in the previous sections. We hope that the following material will be used as new study areas:

– The first general note would be as follows. The studies here can be thought of as the initial step of a general idea, namely constructing a new method (or a test) for the minimality (while the inefficiency holds) of group (in Section 2) and monoid (in Section 3) presentations other than the methods presented in Lemma 1 and Lemma 2, respectively. Especially for the monoid case, although Lemma 2 has not been published, the theory of it has been used widely in last ten years. Therefore, by using generating functions, to obtain a new test on the minimality of monoids would be an interesting and important result.

– As we noted in Remark 1, to study with the minimal presentations has an advantage for our aim in this paper. Conversely, the use of generating functions to obtain a presentation with a minimal number of generators is still an open question.

– Until now, any result to check whether a semigroup presentation is minimal while it is inefficient has not been published. Therefore the whole idea of this paper can also be used for this case.

– It is known that the chemical energy is one of most important application areas of graph theory (cf.[37]). So, as a next step of the expressions in Remark 2, it is worth studying whether this chemical energy can also be obtained from pictures.

– We believe that the same approximation between presentations and generating functions as done in this paper can also be applied to some other special cases of groups and monoids other than <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M208">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M449">View MathML</a>. Moreover, one can investigate which type of polynomials (other than depicted in here) can be used for the general case.

– Here we used exponent sums of pictures as a method to obtain constants of functions. What other methods other than this geometric way can be used could be studied.

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

Dedicated to Professor Hari M Srivastava.

All authors are partially supported by Research Project Offices of Uludağ (2012-15 and 2012-19), Selçuk and Akdeniz Universities.

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