Analysis approach to finite monoids
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Abstract
In a previous paper by the authors, a new approach between algebra and analysis has been recently developed. In detail, it has been generally described how one can express some algebraic properties in terms of special generating functions. To continue the study of this approach, in here, we state and prove that the presentation which has the minimal number of generators of the split extension of two finite monogenic monoids has different sets of generating functions (such that the number of these functions is equal to the number of generators) that represent the exponent sums of the generating pictures of this presentation. This study can be thought of as a mixture of pure analysis, topology and geometry within the purposes of this journal.
AMS Subject Classification:11B68, 11S40, 12D10, 20M05, 20M50, 26C05, 26C10.
Keywords
efficiency pCockcroft property split extension generating functions Stirling numbers array polynomials1 Introduction and preliminaries
Associated with any (connected) topological space X is its fundamental group ${\pi}_{1}(X)$ or 2complex (Squier complex) $\mathcal{D}(X)$. This can often be specified by means of a presentation. A presentation of a group G or monoid M consists of a set of generators of G or M, together with a collection of relations amongst these generators, such that any other relation amongst the generators is derivable (in a precise sense) from the given relations. Algebraic information about ${\pi}_{1}(X)$ or $\mathcal{D}(X)$ can be used to obtain topological information about X (cf. [1]). Many techniques of this branch of mathematics are purely algebraic, and it is possible to achieve much using these techniques. However, in recent years many techniques involving geometric ideas have emerged and are proving more fruitful. These geometric techniques involve graph theory, the theory of tessellations of various surfaces and covering space theory, to name a few.
The number of vertexcolorings of a graph is given by a polynomial on the number of used colors (see [2]). Based on this polynomial, one can define the chromatic number as the minimum number of colors such that the chromatic polynomial is positive. Recently, our attention has been drawn to the paper [3] which is a generalization on the chromatic polynomial of a graph subdivision, and basically the authors determine the chromatic number for a simple graph and then present the generalized polynomial for a particular case of graph subdivision. In this reference, the main idea was to express some graph theoretical parameters in terms of special functions. In a similar manner within algebra, by considering a group or a monoid presentation $\mathcal{P}$, an approximation from algebra to analysis has been recently developed [4]. To do that, the authors supposed $\mathcal{P}$ satisfies the special algebraic properties either efficiency or inefficiency while it is minimal. (The reason for choosing efficiency or (minimal) inefficiency was to have an advantage to work on a minimal number of generators.) Then it was investigated whether some generating functions can be applied, and then it was studied what kind of new properties can be obtained by considering special generating functions over $\mathcal{P}$. In fact, to investigate this theory, $\mathcal{P}$ has been taken as the presentation of the split extension ${\mathbb{Z}}_{n}\u22ca\mathbb{Z}$ and ${\mathbb{Z}}^{2}\u22ca\mathbb{Z}$, respectively. Since the results in [3] imply a new studying area for graphs in the meaning of representation of parameters by generating functions, the results in [4] will be also given an opportunity to make a new classification of infinite groups and monoids by using generating functions.
This paper can be thought of as another version of [4]. Our general aim here is to define some generating functions in terms of the minimality of the given presentation. This will imply that the minimal number of generators can be represented as generating functions. Similarly as in [4], our approximation will be applied by considering the split extension. Here, the split extension will be defined as a semidirect product of two finite monogenic (cyclic) monoids (we may refer to [5] for details on these monoids). It is obvious that the split extension of two finite structures will also be finite. So, the main difference between the results in here and in the paper [4] lies in this fact. Because, while a classification over special cases of infinite groups or monoids was given in [4], the classification in the present paper only focuses on the finite monoids. It is well known that giving some different approximations over finite cases is also as important as giving those over infinite cases.
In the following first subsection, as supportive material, some algebraic facts over split extensions (equivalently, semidirect products), presentations of finite monogenic monoids, a trivializer set of these presentations and efficiency (equivalently, pCockcroft property) are reminded. In Section 2, we present the main material of this paper as two separate subsections. In the first subsection, we present some known results about necessary conditions for the presentation, say ${\mathcal{P}}_{M}$, of the split extension of two finite monogenic monoids to be pCockcroft (see Proposition 2.1 below) and to be minimal but inefficient (see Proposition 2.3 below). In the final subsection, as a result of all theories until there, we introduce generating functions related to our title (see Theorems 2.5, 2.7 and 2.12 below). In Section 3, by considering one of the functions defined in the previous section, we study this function in the meaning of again generating functions and functional equations (see Theorems 3.1 and 3.3 below).
1.1 Fundamentals of the algebraic part
This subsection should be completely thought of as a part of the expressions in the beginning of this paper.
Let $\mathcal{P}=[X;\mathbf{r}]$ be a monoid presentation where a typical element $R\in \mathbf{r}$ has the form ${R}_{+}={R}_{}$. Here ${R}_{+}$, ${R}_{}$ are words on X (that is, elements of the free monoid $F(X)$ on X). The monoid defined by $[X;\mathbf{r}]$ is the quotient of $F(X)$ by the smallest congruence generated by r.
We have a (Squier) graph $\mathrm{\Gamma}=\mathrm{\Gamma}(X;\mathbf{r})$ associated with $[X;\mathbf{r}]$, where the vertices are the elements of $F(X)$ and the edges are the 4tuples $e=(U,R,\epsilon ,V)$, where $U,V\in F(X)$, $R\in \mathbf{r}$ and $\epsilon =\pm 1$. The initial, terminal and inversion functions for an edge e as given above are defined by $\iota (e)=U{R}_{\epsilon}V$, $\tau (e)=U{R}_{\epsilon}V$ and ${e}^{1}=(U,R,\epsilon ,V)$.
Two paths π and ${\pi}^{\mathrm{\prime}}$ in a 2complex are equivalent if there is a finite sequence of paths $\pi ={\pi}_{0},{\pi}_{1},\dots ,{\pi}_{m}={\pi}^{\mathrm{\prime}}$, where for $1\le i\le m$, the path ${\pi}_{i}$ is obtained from ${\pi}_{i1}$ either by inserting or deleting a pair $e{e}^{1}$ of inverse edges or else by inserting or deleting a defining path for one of the 2cells of the complex. There is an equivalence relation, ∼, on paths in Γ which is generated by $({e}_{1}\cdot \iota ({e}_{2}))(\tau ({e}_{1})\cdot {e}_{2})\sim (\iota ({e}_{1})\cdot {e}_{2})({e}_{1}\cdot \tau ({e}_{2}))$ for any edges ${e}_{1}$ and ${e}_{2}$ of Γ. This corresponds to requiring the closed paths $({e}_{1}\cdot \iota ({e}_{2}))(\tau ({e}_{1})\cdot {e}_{2})({e}_{1}^{1}\cdot \tau ({e}_{2}))(\iota ({e}_{1})\cdot {e}_{2}^{1})$ at the vertex $\iota ({e}_{1})\iota ({e}_{2})$ to be the defining paths for the 2cells of a 2complex having Γ as its 1skeleton. This 2complex is called the Squier complex of $\mathcal{P}$ and denoted by $\mathcal{D}(\mathcal{P})$ (see, for example, [6, 7, 8, 9]). The paths in $\mathcal{D}(\mathcal{P})$ can be represented by geometric configurations, called monoid pictures. We assume here that the reader is familiar with monoid pictures (see [[6], Section 4], [[7], Section 1] or [[8], Section 2]). Typically, we will use blackboard bold, e.g., $\mathbb{A}$, $\mathbb{B}$, ℂ, ℙ, as notation for monoid pictures. Atomic monoid pictures are pictures which correspond to paths of length 1. Write $[U,R,\epsilon ,V]$ for the atomic picture which corresponds to the edge $(U,R,\epsilon ,V)$ of the Squier complex. Whenever we can concatenate two paths π and ${\pi}^{\mathrm{\prime}}$ in Γ to form the path $\pi {\pi}^{\mathrm{\prime}}$, then we can concatenate the corresponding monoid pictures ℙ and ${\mathbb{P}}^{\mathrm{\prime}}$ to form a monoid picture $\mathbb{P}{\mathbb{P}}^{\mathrm{\prime}}$ corresponding to $\pi {\pi}^{\mathrm{\prime}}$. The equivalence of paths in the Squier complex corresponds to an equivalence of monoid pictures. That is, two monoid pictures ℙ and ${\mathbb{P}}^{\mathrm{\prime}}$ are equivalent if there is a finite sequence of monoid pictures $\mathbb{P}={\mathbb{P}}_{0},{\mathbb{P}}_{1},\dots ,{\mathbb{P}}_{m}={\mathbb{P}}^{\mathrm{\prime}}$ where, for $1\le i\le m$, the monoid picture ${\mathbb{P}}_{i}$ is obtained from the picture ${\mathbb{P}}_{i1}$ either by inserting or deleting a subpicture $\mathbb{A}{\mathbb{A}}^{1}$, where $\mathbb{A}$ is an atomic monoid picture, or else by replacing a subpicture $(\mathbb{A}\cdot \iota (\mathbb{B}))(\tau (\mathbb{A})\cdot \mathbb{B})$ by $(\iota (\mathbb{A})\cdot \mathbb{B})(\mathbb{A}\cdot \tau (\mathbb{B}))$ or vice versa, where $\mathbb{A}$ and $\mathbb{B}$ are atomic monoid pictures.
A monoid picture is called a spherical monoid picture when the corresponding path in the Squier complex is a closed path. Suppose Y is a collection of spherical monoid pictures over $\mathcal{P}$. Two monoid pictures ℙ and ${\mathbb{P}}^{\mathrm{\prime}}$ are equivalent relative to Y if there is a finite sequence of monoid pictures $\mathbb{P}={\mathbb{P}}_{0},{\mathbb{P}}_{1},\dots ,{\mathbb{P}}_{m}={\mathbb{P}}^{\mathrm{\prime}}$ where, for $1\le i\le m$, the monoid picture ${\mathbb{P}}_{i}$ is obtained from the picture ${\mathbb{P}}_{i1}$ either by the insertion, deletion and replacement operations of the previous paragraph or else by inserting or deleting a subpicture of the form $W\cdot \mathbb{Y}\cdot V$ or of the form $W\cdot {\mathbb{Y}}^{1}\cdot V$, where $W,V\in F(X)$ and $\mathbb{Y}\in \mathbf{Y}$. By definition, a set Y of spherical monoid pictures over $\mathcal{P}$ is a trivializer of $\mathcal{D}(\mathcal{P})$ if every spherical monoid picture is equivalent to an empty picture relative to Y. By [[7], Theorem 5.1], if Y is a trivializer for the Squier complex, then the elements of Y generate the first homology group of the Squier complex. The trivializer is also called a set of generating pictures. Some examples and more details of the trivializer can be found in [7, 8, 9, 10, 11, 12, 13, 14].
For any monoid picture ℙ over $\mathcal{P}$ and for any $R\in \mathbf{r}$, ${exp}_{R}(\mathbb{P})$ denotes the exponent sum of R in ℙ which is the number of positive discs labeled by ${R}_{+}$, minus the number of negative discs labeled by ${R}_{}$. For a nonnegative integer n, $\mathcal{P}$ is said to be nCockcroft if ${exp}_{R}(\mathbb{P})\equiv 0$ (modn), (where congruence (mod0) is taken to be equality) for all $R\in \mathbf{r}$ and for all spherical pictures ℙ over $\mathcal{P}$. Then a monoid ℳ is said to be nCockcroft if it admits an nCockcroft presentation. In fact, to verify the nCockcroft property, it is enough to check for pictures $\mathbb{P}\in \mathbf{Y}$, where Y is a trivializer (see [7, 8]). The 0Cockcroft property is usually just called Cockcroft. In general, we take n to be equal to 0 or a prime p. Examples of monoid presentations with Cockcroft and pCockcroft properties can be found in [10].
Suppose that $\mathcal{P}=[X;\mathbf{r}]$ is a finite presentation for a monoid ℳ. Then the Euler characteristic $\chi (\mathcal{P})$ is defined by $\chi (\mathcal{P})=1X+\mathbf{r}$ and $\delta (\mathcal{M})$ is defined by $\delta (\mathcal{M})=1r{k}_{\mathbb{Z}}({H}_{1}(\mathcal{M}))+d({H}_{2}(\mathcal{M}))$. In unpublished work, Pride has shown that $\chi (\mathcal{P})\ge \delta (\mathcal{M})$. With this background, we define the finite monoid presentation $\mathcal{P}$ to be efficient if $\chi (\mathcal{P})=\delta (\mathcal{M})$, and we define the monoid ℳ to be efficient if it has an efficient presentation. Moreover, a presentation ${\mathcal{P}}_{0}$ for ℳ is called minimal if $\chi ({\mathcal{P}}_{0})\le \chi (\mathcal{P})$ for all presentations $\mathcal{P}$ of ℳ. There is also interest in finding inefficient finitely presented monoids since if we can find a minimal presentation ${\mathcal{P}}_{0}$ for a monoid ℳ such that ${\mathcal{P}}_{0}$ is not efficient, then we have $\chi ({\mathcal{P}}^{\mathrm{\prime}})\ge \chi ({\mathcal{P}}_{0})>\delta (\mathcal{M})$ for all presentations ${\mathcal{P}}^{\mathrm{\prime}}$ defining the same monoid ℳ. Thus, there is no efficient presentation for ℳ, that is, ℳ is not an efficient monoid.
The following theorem was first given in [10]. (The group version of this result was proved by Epstein in [15].)
Theorem 1.1 Let $\mathcal{P}$ be a monoid presentation. Then $\mathcal{P}$ is efficient if and only if it is pCockcroft for some prime p.
Let ${I}_{2}^{(l)}(\mathcal{P})$ be a twosided ideal of $\mathbb{Z}\mathcal{M}$ generated by the elements ${\lambda}_{\mathbb{P},S}$, where ℙ is a spherical monoid picture and $S\in \mathbf{r}$. Then this ideal is called the second Fox ideal of $\mathcal{P}$. More specifically, for a trivializer Y of $\mathcal{D}(\mathcal{P})$, the set ${I}_{2}^{(l)}(\mathcal{P})$ is generated (as twosided ideal) by the elements ${\lambda}_{\mathbb{P},S}$, where $\mathbb{P}\in \mathbf{Y}$ and $S\in \mathbf{r}$. We note that all this above material given by the consideration ‘left’ can also be applied to ‘right’ for a monoid ℳ.
is a standard monoid presentation for the semidirect product ℳ.
In [14], a finite trivializer set has been constructed for the standard presentation ${\mathcal{P}}_{\mathcal{M}}$, as given in (3), for the semidirect product ℳ. We will essentially follow [10] in describing this trivializer set using spherical pictures and certain nonspherical subpictures of these.
If $W={y}_{1}{y}_{2}\cdots {y}_{m}$ is a positive word on Y, then for any $x\in X$, we denote the word $({y}_{1}{\theta}_{x})({y}_{2}{\theta}_{x})\cdots ({y}_{m}{\theta}_{x})$ by $W{\theta}_{x}$. If $U={x}_{1}{x}_{2}\cdots {x}_{n}$ is a positive word on X, then for any $y\in Y$, we denote the word $(\cdots ((y{\theta}_{{x}_{1}}){\theta}_{{x}_{2}}){\theta}_{{x}_{3}}\cdots ){\theta}_{{x}_{n}}$ by $y{\theta}_{U}$, and this can be represented by a monoid picture, say ${\mathbb{A}}_{U,y}$, as in Figure 2(b). For $y\in Y$ and the relation ${R}_{+}={R}_{}$ in the relation set r, we have two important special cases, ${\mathbb{A}}_{{R}_{+},y}$ and ${\mathbb{A}}_{{R}_{},y}$, of this consideration. We should note that these nonspherical pictures consist of only ${T}_{yx}$discs ($x\in X$). Let $S\in \mathbf{s}$ and $x\in X$. Since ${[{S}_{+}{\theta}_{x}]}_{{\mathcal{P}}_{K}}={[{S}_{}{\theta}_{x}]}_{{\mathcal{P}}_{K}}$, there is a nonspherical picture, say ${\mathbb{B}}_{S,x}$, over ${\mathcal{P}}_{K}$ with $\iota ({\mathbb{B}}_{S,x})={S}_{+}{\theta}_{x}$ and $\tau ({\mathbb{B}}_{S,x})={S}_{}{\theta}_{x}$. Further, let ${R}_{+}={R}_{}$ be a relation $R\in \mathbf{r}$ and $y\in Y$. Since θ is a homomorphism, by the definition on $y{\theta}_{U}$, we have that $y{\theta}_{{R}_{+}}$ and $y{\theta}_{{R}_{}}$ must represent the same element of the monoid K. That is, ${[y{\theta}_{{R}_{+}}]}_{{\mathcal{P}}_{K}}={[y{\theta}_{{R}_{}}]}_{{\mathcal{P}}_{K}}$. Hence, there is a nonspherical picture over ${\mathcal{P}}_{K}$ which we denote by ${\mathbb{C}}_{y,{\theta}_{R}}$ with $\iota ({\mathbb{C}}_{y,{\theta}_{R}})=y{\theta}_{{R}_{+}}$ and $\tau ({\mathbb{C}}_{y,{\theta}_{R}})=y{\theta}_{{R}_{}}$. In fact, there may be many different ways to construct the pictures ${\mathbb{B}}_{S,x}$ and ${\mathbb{C}}_{y,{\theta}_{R}}$. These pictures must exist, but they are not unique. On the other hand, the picture ${\mathbb{A}}_{U,y}$ will depend upon our choices for words $y{\theta}_{x}$, but this is unique once these choices are made.
After all, for $x\in X$, $y\in Y$, $R\in \mathbf{r}$ and $S\in \mathbf{s}$, one can construct spherical monoid pictures, say ${\mathbb{P}}_{S,x}$ and ${\mathbb{P}}_{R,y}$, by using the nonspherical pictures ${\mathbb{B}}_{S,x}$, ${\mathbb{A}}_{{R}_{+},y}$, ${\mathbb{A}}_{{R}_{},y}$ and ${\mathbb{C}}_{y,{\theta}_{R}}$ (see Figures 2, 3 and 4 for the examples of these pictures). Let ${\mathbf{X}}_{\mathbf{A}}$ and ${\mathbf{X}}_{\mathbf{K}}$ be trivializer sets of $\mathcal{D}({\mathcal{P}}_{A})$ and $\mathcal{D}({\mathcal{P}}_{K})$, respectively. Also, let ${\mathbf{C}}_{\mathbf{1}}=\{{\mathbb{P}}_{S,x}:S\in \mathbf{s},x\in X\}$ and ${\mathbf{C}}_{\mathbf{2}}=\{{\mathbb{P}}_{R,y}:R\in \mathbf{r},y\in Y\}$. Then, by [10, 14], it is known that for a presentation ${\mathcal{P}}_{\mathcal{M}}$, as in (3), a trivializer set of $\mathcal{D}({\mathcal{P}}_{\mathcal{M}})$ is ${\mathbf{X}}_{\mathcal{M}}={\mathbf{X}}_{\mathbf{A}}\cup {\mathbf{X}}_{\mathbf{K}}\cup {\mathbf{C}}_{\mathbf{1}}\cup {\mathbf{C}}_{\mathbf{2}}$.
2 Generators over the semidirect product of finite cyclic monoids
In fact, this is the main section of the paper and it will be given as two subsections under the names of Part I and Part II. Since we will define generating functions by considering the exponent sums of the generating pictures over the presentation of this semidirect product, the first subsection is aimed to define these generating pictures and the related results about them.
2.1 Part I: generating pictures
In this subsection, we will mainly present the efficiency (equivalently, pCockcroft property for a prime p by Theorem 1.1) for the semidirect products of finite cyclic monoids.
In the rest of the paper, we will assume that the equality in Equation (6) holds when we talk about the semidirect product M of K by A.
The following result states necessary and sufficient conditions for the presentation of the split extension of two finite monogenic monoids to be efficient.
Proposition 2.1 ([18])
Remark 2.2 To be an example of Proposition 2.1, one can take

$k=10$, $l=6$, $\mu =4$, $\lambda =2$ or $k=6$, $l=2$, $\mu =5$, $\lambda =3$ while $p=2$ and $i=3$ to get 2Cockcroft property for the presentation ${\mathcal{P}}_{M}$ in (7), or more generally

for any prime p, $k=(p+1)[\frac{{(p+1)}^{p}1}{p}]+1$, $l=1$, $\mu =p+1=i$ and $\lambda =1$ to get pCockcroft property for the presentation ${\mathcal{P}}_{M}$ in (7).
In particular, if we choose ${exp}_{S}({\mathbb{B}}_{S,x})=i=0$ or 2, then ${\mathcal{P}}_{M}$ will be inefficient.
Recall that, by the meaning of finite cyclic monoids, ${exp}_{y}(S)=kl$ cannot be equal to 0. We also note that a similar proof for the following result about minimal but inefficiency of ${\mathcal{P}}_{M}$ can be found in [18].
Proposition 2.3 Let M be the semidirect product of K by A, and let ${\mathcal{P}}_{M}$, as in (7), be the presentation for M where $l,k,\lambda ,\mu ,i\in {\mathbb{Z}}^{+}$ and $l<k$, $\lambda <\mu $. If $i=2$ and the subtraction $kl$ is not even and not equal to 1, then ${\mathcal{P}}_{M}$ is minimal but inefficient.
Remark 2.4 To be an example of Proposition 2.3, we can consider the following:

For an odd positive integer t, one can take $k=4t$, $l=t$, $\mu =3t$, $\lambda =t$ and $i=2$ in the presentation given in (7). Since $kl=3t\ne 2n$ ($n\in {\mathbb{Z}}^{+}$), the presentation is minimal but inefficient.

For all $s,t\in {\mathbb{Z}}^{+}$ such that $s<t$, one can take $k=2t+1$, $l=2s$, $\mu =kl$, $\lambda =1$ and $i=2$ in the presentation given in (7). Since $kl=2(ts)+1\ne 2n$ ($n\in {\mathbb{Z}}^{+}$), the presentation is minimal while it is inefficient.
2.2 Part II: generating functions
By considering the pictures defined in the previous section and also the evaluations obtained from them, we will define the related generating functions. In another words, by taking into account Propositions 2.1 and 2.3, we will reach our main aim over monoids of this paper.
We firstly recall that, as noted in [[4], Remark 1.1], if a monoid presentation satisfies efficiency or inefficiency (while it is minimal), then it always has a minimal number of generators. Working with the minimal number of elements gives a great opportunity to define related generating functions over this presentation. This will be one of the key points in our results.
Since the coefficients of array polynomials are integers, they find very large application area, especially in system control (cf. [22]). In fact, these integer coefficients give us the opportunity to use these polynomials in our case. We should note that there also exist some other polynomials, namely Dickson, Bell, Abel, MittagLeffler etc., which have integer coefficients which will not be handled in this paper.
From (5), we know that $\mu =\lambda +r$, where $1\le r\le \mu 1$. Hence, by considering the meaning and conditions of Proposition 2.1, we obtain the following theorem as one of the main results of this paper.
where ${S}_{a}^{n}(x)$ and ${S}_{a}^{n}(y)$ are defined as in (8).
Proof Let us consider the generating pictures ${\mathbb{P}}_{S,x}$, ${\mathbb{P}}_{R,y}$ (in Figure 4) with their nonspherical subpictures defined in Figures 2 and 3, and the generating pictures of finite monogenic monoids defined in Figure 1. Recall that by counting the exponent sums of the discs R, S and ${T}_{yx}$ in the related pictures, the conditions of Proposition 2.1 have been obtained [18]. (For more similar results and applications, one can see the papers [10, 11].)
Then, by reformulating the elements in (10) and (11) of the second Fox ideal ${I}_{2}^{(l)}({\mathcal{P}}_{M})$, we arrive at the functions in (9) as desired. □
Considering Remark 2.2, we obtain the following corollary as a consequence of Theorem 2.5.
In Proposition 2.3, the minimality (while satisfying inefficiency) of the presentation ${\mathcal{P}}_{M}$ was expressed in (7). Thus, by considering the meaning and conditions of Proposition 2.3, we obtain the following theorem as another main result of this paper. Since the proof is quite similar to the proof of Theorem 2.5, we omit it.
where $kl$ is an odd integer and ${S}_{a}^{n}(x)$ and ${S}_{a}^{n}(y)$ are defined as in (8).
By considering Remark 2.4, we can have the following consequences of Theorem 2.7.
Remark 2.10 Since both presentations in Propositions 2.1 and 2.3 have the minimal number of generators because of their efficiency or inefficiency (but minimal) status, this situation affected very positively the number and type of generating functions defined on them.
At this point, we should note that for ${t}_{1}\ne {t}_{2}\in {\mathbb{R}}^{+}$, $\gamma \in \mathbb{C}$, $a\in {\mathbb{N}}_{0}$, generalized array type polynomials ${\mathcal{S}}_{a}^{n}(x;{t}_{1},{t}_{2};\gamma )$ related to the nonnegative real parameters have been recently developed (in [20]) and some elementary properties including recurrence relations of these polynomials have been derived. In fact, by setting ${t}_{1}=\gamma =1$ and ${t}_{2}=e$, Equation (8) is obtained.
Remark 2.11 For a future project, one can study the generalization of Theorems 2.5 and 2.7 by using ${\mathcal{S}}_{a}^{n}(x;{t}_{1},{t}_{2};\gamma )$.
where ${\delta}_{n,0}$ denotes the Kronecker symbol (see [27, 28]). It is known that Stirling numbers are used in combinatorics, in number theory, in discrete probability distributions for finding higherorder moments, etc. We finally note that since $S(n,a)$ is the number of ways to partition a set of n objects into k groups, these numbers find an application area in combinatorics and in theory of partitions.
After that, as a different version of Theorem 2.5 (and so Theorem 2.7), we present the following result.
Theorem 2.12 The efficient presentation ${\mathcal{P}}_{M}$ in (7) has a set of generating functions in terms of Stirling numbers as given in (13). By taking $i=2$ and $kl$ is an odd positive integer, we get a set of generating functions in terms of Stirling numbers for the inefficient but minimal presentation of the form as defined in (7).
(see [27, 28]). According to the same references, by substituting $\gamma =1$ into the above equation, the Stirling numbers of the second kind $S(n,v)$ are obtained.
Hence, we can present the following note.
Remark 2.13 It is clearly seen that Stirling numbers have been only considered in Theorems 2.5 and 2.7 (and the corollaries about them). However, one can also study the γStirling numbers $S(n,v;\gamma )$ defined in (16) and generalized γStirling numbers $\mathcal{S}(n,v;a,b;\gamma )$ defined in (15) to obtain different types of generating functions.
3 The constant function related to main results
as the constants of defined generating functions. In this section, by representing these constants as a single function (see Equation (17) below), we investigate some new properties over it.
which is actually mtimes algebraic multiplication of the function f.
All these above processes imply the following result.
and $S(a,m)$ denotes the Stirling numbers of second kind.
Some properties of the function $Y(z)={[f(z,r,\lambda ,k,l)]}^{m}$ in (19) can be expressed as follows:

If $z\in \mathbb{C}$, then $Y(z)$ is an analytic function, and then it has a power series as defined in the above theorem with Equation (22).

If $z\in \mathbb{R}$, then $Y(z)$ is a continuous function which is actually a polynomial function having degree $\lambda +r$.

If we replace z by ${e}^{t}$ in (19), we obtain the second kind Stirling numbers.
since $S(n,1)=1$.
By considering [[20], Eq. (3.2)] and Equation (20), we can extend Remark 3.2 to a general natural number $m>1$ as in the following theorem.
where ${\mathcal{S}}_{m}^{n}(m\lambda )$ denotes the array polynomials.
As it was seen, only the function defined in (18) itself is enough to represent almost all the conditions in Propositions 2.1 and 2.3. Thus, we can express the following remark which depicts some new studying areas for a future project.
Remark 3.4

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

While $z\in \mathbb{C}$ and $z\in \mathbb{R}$, analytic and functional equations can be studied.

As we have partially done in the above, replacing z by ${e}^{t}$, one can study the generating functions of array polynomials and Stirling numbers.
3.1 Some other properties over this constant
where $\mathcal{A}$ stands for some constants.
This above theory is related to the functional equations. In fact, these above progresses show that the presentation ${\mathcal{P}}_{M}$ in (7) can be related to the functional equations.
Notes
Acknowledgements
All authors are partially supported by Research Project Offices of Uludağ, Selçuk and Akdeniz Universities, and TUBITAK (The Scientific and Technological Research Council of Turkey).
Supplementary material
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