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Analysis approach to finite monoids

  • A Sinan ÇevikEmail author
  • I Naci Cangül
  • Yılmaz Şimşek
Open Access
Research
Part of the following topical collections:
  1. Proceedings of the International Congress in Honour of Professor Hari M. Srivastava

Abstract

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

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

Keywords

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

1 Introduction and preliminaries

Associated with any (connected) topological space X is its fundamental group π 1 ( X ) Open image in new window or 2-complex (Squier complex) D ( X ) Open image in new window. 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 π 1 ( X ) Open image in new window or D ( X ) Open image in new window can be used to obtain topological information about X (cf. [1]). Many techniques of this branch of mathematics are purely algebraic, and it is possible to achieve much using these techniques. However, in recent years many techniques involving geometric ideas have emerged and are proving more fruitful. These geometric techniques involve graph theory, the theory of tessellations of various surfaces and covering space theory, to name a few.

The number of vertex-colorings of a graph is given by a polynomial on the number of used colors (see [2]). Based on this polynomial, one can define the chromatic number as the minimum number of colors such that the chromatic polynomial is positive. Recently, our attention has been drawn to the paper [3] which is a generalization on the chromatic polynomial of a graph subdivision, and basically the authors determine the chromatic number for a simple graph and then present the generalized polynomial for a particular case of graph subdivision. In this reference, the main idea was to express some graph theoretical parameters in terms of special functions. In a similar manner within algebra, by considering a group or a monoid presentation P Open image in new window, an approximation from algebra to analysis has been recently developed [4]. To do that, the authors supposed P Open image in new window 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 P Open image in new window. In fact, to investigate this theory, P Open image in new window has been taken as the presentation of the split extension Z n Z Open image in new window and Z 2 Z Open image in new window, respectively. Since the results in [3] imply a new studying area for graphs in the meaning of representation of parameters by generating functions, the results in [4] will be also given an opportunity to make a new classification of infinite groups and monoids by using generating functions.

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

In the following first subsection, as supportive material, some algebraic facts over split extensions (equivalently, semi-direct products), presentations of finite monogenic monoids, a trivializer set of these presentations and efficiency (equivalently, p-Cockcroft property) are reminded. In Section 2, we present the main material of this paper as two separate subsections. In the first subsection, we present some known results about necessary conditions for the presentation, say P M Open image in new window, of the split extension of two finite monogenic monoids to be p-Cockcroft (see Proposition 2.1 below) and to be minimal but inefficient (see Proposition 2.3 below). In the final subsection, as a result of all theories until there, we introduce generating functions related to our title (see Theorems 2.5, 2.7 and 2.12 below). In Section 3, by considering one of the functions defined in the previous section, we study this function in the meaning of again generating functions and functional equations (see Theorems 3.1 and 3.3 below).

1.1 Fundamentals of the algebraic part

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

Let P = [ X ; r ] Open image in new window be a monoid presentation where a typical element R r Open image in new window has the form R + = R Open image in new window. Here R + Open image in new window, R Open image in new window are words on X (that is, elements of the free monoid F ( X ) Open image in new window on X). The monoid defined by [ X ; r ] Open image in new window is the quotient of F ( X ) Open image in new window by the smallest congruence generated by r.

We have a (Squier) graph Γ = Γ ( X ; r ) Open image in new window associated with [ X ; r ] Open image in new window, where the vertices are the elements of F ( X ) Open image in new window and the edges are the 4-tuples e = ( U , R , ε , V ) Open image in new window, where U , V F ( X ) Open image in new window, R r Open image in new window and ε = ± 1 Open image in new window. The initial, terminal and inversion functions for an edge e as given above are defined by ι ( e ) = U R ε V Open image in new window, τ ( e ) = U R ε V Open image in new window and e 1 = ( U , R , ε , V ) Open image in new window.

Two paths π and π Open image in new window in a 2-complex are equivalent if there is a finite sequence of paths π = π 0 , π 1 , , π m = π Open image in new window, where for 1 i m Open image in new window, the path π i Open image in new window is obtained from π i 1 Open image in new window either by inserting or deleting a pair e e 1 Open image in new window of inverse edges or else by inserting or deleting a defining path for one of the 2-cells of the complex. There is an equivalence relation, ∼, on paths in Γ which is generated by ( e 1 ι ( e 2 ) ) ( τ ( e 1 ) e 2 ) ( ι ( e 1 ) e 2 ) ( e 1 τ ( e 2 ) ) Open image in new window for any edges e 1 Open image in new window and e 2 Open image in new window of Γ. This corresponds to requiring the closed paths ( e 1 ι ( e 2 ) ) ( τ ( e 1 ) e 2 ) ( e 1 1 τ ( e 2 ) ) ( ι ( e 1 ) e 2 1 ) Open image in new window at the vertex ι ( e 1 ) ι ( e 2 ) Open image in new window to be the defining paths for the 2-cells of a 2-complex having Γ as its 1-skeleton. This 2-complex is called the Squier complex of P Open image in new window and denoted by D ( P ) Open image in new window (see, for example, [6, 7, 8, 9]). The paths in D ( P ) Open image in new window can be represented by geometric configurations, called monoid pictures. We assume here that the reader is familiar with monoid pictures (see [[6], Section 4], [[7], Section 1] or [[8], Section 2]). Typically, we will use blackboard bold, e.g., A Open image in new window, B Open image in new window, ℂ, ℙ, as notation for monoid pictures. Atomic monoid pictures are pictures which correspond to paths of length 1. Write [ | U , R , ε , V | ] Open image in new window for the atomic picture which corresponds to the edge ( U , R , ε , V ) Open image in new window of the Squier complex. Whenever we can concatenate two paths π and π Open image in new window in Γ to form the path π π Open image in new window, then we can concatenate the corresponding monoid pictures ℙ and P Open image in new window to form a monoid picture P P Open image in new window corresponding to π π Open image in new window. The equivalence of paths in the Squier complex corresponds to an equivalence of monoid pictures. That is, two monoid pictures ℙ and P Open image in new window are equivalent if there is a finite sequence of monoid pictures P = P 0 , P 1 , , P m = P Open image in new window where, for 1 i m Open image in new window, the monoid picture P i Open image in new window is obtained from the picture P i 1 Open image in new window either by inserting or deleting a subpicture A A 1 Open image in new window, where A Open image in new window is an atomic monoid picture, or else by replacing a subpicture ( A ι ( B ) ) ( τ ( A ) B ) Open image in new window by ( ι ( A ) B ) ( A τ ( B ) ) Open image in new window or vice versa, where A Open image in new window and B Open image in new window 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 P Open image in new window. Two monoid pictures ℙ and P Open image in new window are equivalent relative to Y if there is a finite sequence of monoid pictures P = P 0 , P 1 , , P m = P Open image in new window where, for 1 i m Open image in new window, the monoid picture P i Open image in new window is obtained from the picture P i 1 Open image in new window either by the insertion, deletion and replacement operations of the previous paragraph or else by inserting or deleting a subpicture of the form W Y V Open image in new window or of the form W Y 1 V Open image in new window, where W , V F ( X ) Open image in new window and Y Y Open image in new window. By definition, a set Y of spherical monoid pictures over P Open image in new window is a trivializer of D ( P ) Open image in new window 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 P Open image in new window and for any R r Open image in new window, exp R ( P ) Open image in new window denotes the exponent sum of R in ℙ which is the number of positive discs labeled by R + Open image in new window, minus the number of negative discs labeled by R Open image in new window. For a non-negative integer n, P Open image in new window is said to be n-Cockcroft if exp R ( P ) 0 Open image in new window (modn), (where congruence (mod0) is taken to be equality) for all R r Open image in new window and for all spherical pictures ℙ over P Open image in new window. Then a monoid ℳ is said to be n-Cockcroft if it admits an n-Cockcroft presentation. In fact, to verify the n-Cockcroft property, it is enough to check for pictures P Y Open image in new window, where Y is a trivializer (see [7, 8]). The 0-Cockcroft property is usually just called Cockcroft. In general, we take n to be equal to 0 or a prime p. Examples of monoid presentations with Cockcroft and p-Cockcroft properties can be found in [10].

Suppose that P = [ X ; r ] Open image in new window is a finite presentation for a monoid ℳ. Then the Euler characteristic χ ( P ) Open image in new window is defined by χ ( P ) = 1 | X | + | r | Open image in new window and δ ( M ) Open image in new window is defined by δ ( M ) = 1 r k Z ( H 1 ( M ) ) + d ( H 2 ( M ) ) Open image in new window. In unpublished work, Pride has shown that χ ( P ) δ ( M ) Open image in new window. With this background, we define the finite monoid presentation P Open image in new window to be efficient if χ ( P ) = δ ( M ) Open image in new window, and we define the monoid ℳ to be efficient if it has an efficient presentation. Moreover, a presentation P 0 Open image in new window for ℳ is called minimal if χ ( P 0 ) χ ( P ) Open image in new window for all presentations P Open image in new window of ℳ. There is also interest in finding inefficient finitely presented monoids since if we can find a minimal presentation P 0 Open image in new window for a monoid ℳ such that P 0 Open image in new window is not efficient, then we have χ ( P ) χ ( P 0 ) > δ ( M ) Open image in new window for all presentations P Open image in new window 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 P Open image in new window be a monoid presentation. Then P Open image in new window is efficient if and only if it is p-Cockcroft for some prime p.

Let ℳ be a monoid with the presentation P = [ X ; r ] Open image in new window, and let
P ( l ) = S r Z M e S Open image in new window
be the free left Z M Open image in new window-module with basis { e S : S r } Open image in new window. For an atomic picture A = ( U , S , ε , V ) Open image in new window (where U , V F ( x ) Open image in new window, S r Open image in new window, ε = ± 1 Open image in new window), we define eval ( l ) ( A ) = ε U ¯ e S P ( l ) Open image in new window, where U ¯ M ( P ) Open image in new window. For any spherical monoid picture ℙ, we then define
eval ( l ) ( P ) = i = 1 n eval ( l ) ( A i ) P ( l ) . Open image in new window
(1)
Let λ P , S Open image in new window be the coefficient of e S Open image in new window in eval ( l ) ( P ) Open image in new window. So, we can write
eval ( l ) ( P ) = S r λ P , S e S P ( l ) . Open image in new window
(2)

Let I 2 ( l ) ( P ) Open image in new window be a two-sided ideal of Z M Open image in new window generated by the elements λ P , S Open image in new window, where ℙ is a spherical monoid picture and S r Open image in new window. Then this ideal is called the second Fox ideal of P Open image in new window. More specifically, for a trivializer Y of D ( P ) Open image in new window, the set I 2 ( l ) ( P ) Open image in new window is generated (as two-sided ideal) by the elements λ P , S Open image in new window, where P Y Open image in new window and S r Open image in new window. We note that all this above material given by the consideration ‘left’ can also be applied to ‘right’ for a monoid ℳ.

The definition and a standard presentation for the semi-direct product of two monoids can be found in [10, 11, 14, 16]. Let A and K be arbitrary monoids with associated presentations P A = [ X ; r ] Open image in new window and P K = [ Y ; s ] Open image in new window, respectively. Let M = K θ A Open image in new window be the corresponding semi-direct product of these two monoids, where θ is a monoid homomorphism from A to End ( K ) Open image in new window. (We note that the reader can find some examples of monoid endomorphisms in [17].) The elements of ℳ can be regarded as ordered pairs ( a , k ) Open image in new window where a A Open image in new window, k K Open image in new window with multiplication given by ( a , k ) ( a , k ) = ( a a , ( k θ a ) k ) Open image in new window. The monoids A and K are identified with the submonoids of ℳ having elements ( a , 1 ) Open image in new window and ( 1 , k ) Open image in new window, respectively. We want to define standard presentations for ℳ. For every x X Open image in new window and y Y Open image in new window, choose a word, which we denote by y θ x Open image in new window, on Y such that [ y θ x ] = [ y ] θ [ x ] Open image in new window as an element of K. To establish notation, let us denote the relation y x = x ( y θ x ) Open image in new window on X Y Open image in new window by T y x Open image in new window and write t for the set of relations T y x Open image in new window. Then, for any choice of the words y θ x Open image in new window,
P M = [ Y , X ; s , r , t ] Open image in new window
(3)

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

In [14], a finite trivializer set has been constructed for the standard presentation P M Open image in new window, as given in (3), for the semi-direct product ℳ. We will essentially follow [10] in describing this trivializer set using spherical pictures and certain non-spherical subpictures of these.

If W = y 1 y 2 y m Open image in new window is a positive word on Y, then for any x X Open image in new window, we denote the word ( y 1 θ x ) ( y 2 θ x ) ( y m θ x ) Open image in new window by W θ x Open image in new window. If U = x 1 x 2 x n Open image in new window is a positive word on X, then for any y Y Open image in new window, we denote the word ( ( ( y θ x 1 ) θ x 2 ) θ x 3 ) θ x n Open image in new window by y θ U Open image in new window, and this can be represented by a monoid picture, say A U , y Open image in new window, as in Figure 2(b). For y Y Open image in new window and the relation R + = R Open image in new window in the relation set r, we have two important special cases, A R + , y Open image in new window and A R , y Open image in new window, of this consideration. We should note that these non-spherical pictures consist of only T y x Open image in new window-discs ( x X Open image in new window). Let S s Open image in new window and x X Open image in new window. Since [ S + θ x ] P K = [ S θ x ] P K Open image in new window, there is a non-spherical picture, say B S , x Open image in new window, over P K Open image in new window with ι ( B S , x ) = S + θ x Open image in new window and τ ( B S , x ) = S θ x Open image in new window. Further, let R + = R Open image in new window be a relation R r Open image in new window and y Y Open image in new window. Since θ is a homomorphism, by the definition on y θ U Open image in new window, we have that y θ R + Open image in new window and y θ R Open image in new window must represent the same element of the monoid K. That is, [ y θ R + ] P K = [ y θ R ] P K Open image in new window. Hence, there is a non-spherical picture over P K Open image in new window which we denote by C y , θ R Open image in new window with ι ( C y , θ R ) = y θ R + Open image in new window and τ ( C y , θ R ) = y θ R Open image in new window. In fact, there may be many different ways to construct the pictures B S , x Open image in new window and C y , θ R Open image in new window. These pictures must exist, but they are not unique. On the other hand, the picture A U , y Open image in new window will depend upon our choices for words y θ x Open image in new window, but this is unique once these choices are made.

After all, for x X Open image in new window, y Y Open image in new window, R r Open image in new window and S s Open image in new window, one can construct spherical monoid pictures, say P S , x Open image in new window and P R , y Open image in new window, by using the non-spherical pictures B S , x Open image in new window, A R + , y Open image in new window, A R , y Open image in new window and C y , θ R Open image in new window (see Figures 2, 3 and 4 for the examples of these pictures). Let X A Open image in new window and X K Open image in new window be trivializer sets of D ( P A ) Open image in new window and D ( P K ) Open image in new window, respectively. Also, let C 1 = { P S , x : S s , x X } Open image in new window and C 2 = { P R , y : R r , y Y } Open image in new window. Then, by [10, 14], it is known that for a presentation P M Open image in new window, as in (3), a trivializer set of D ( P M ) Open image in new window is X M = X A X K C 1 C 2 Open image in new window.

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

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

2.1 Part I: generating pictures

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

Let A and K be two finite cyclic monoids with presentations
P A = [ x ; x μ = x λ ] and P K = [ y ; y k = y l ] Open image in new window
(4)
respectively, where l , k , λ , μ Z + Open image in new window such that l < k Open image in new window and λ < μ Open image in new window, or equivalently,
μ = λ + r ( 1 r μ 1 ) and k = l + ω ( 1 ω k 1 ) . Open image in new window
(5)
Due to [10], a trivializer set X K Open image in new window (and similarly X A Open image in new window) of the Squier complex D ( P K ) Open image in new window (and similarly D ( P A ) Open image in new window) is given by the pictures P k , l m Open image in new window ( 1 m k 1 Open image in new window), as in Figure 1.
Figure 1

Generating pictures of finite monogenic monoids.

Let ψ i Open image in new window ( 0 i k 1 Open image in new window) be an endomorphism of K. Then we have a mapping x End ( K ) Open image in new window, x ψ i Open image in new window. In fact this induces a homomorphism θ : A End ( K ) Open image in new window, x ψ i Open image in new window if and only if ψ i μ = ψ i λ Open image in new window. Since ψ i μ Open image in new window and ψ i λ Open image in new window are equal if and only if they agree on the generator y of K, we must have
[ y i μ ] = [ y i λ ] . Open image in new window
(6)
We then have the semi-direct product M = K θ A Open image in new window and, by [10], a standard presentation
P M = [ y , x ; S , R , T y x ] , Open image in new window
(7)
as in (3), for the monoid M where
S : y k = y l , R : x μ = x λ and T y x : y x = x y i . Open image in new window

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

The subpicture B S , x Open image in new window can be drawn as in Figure 2(a), and in fact, by considering this subpicture, we clearly have
exp S ( B S , x ) = i . Open image in new window
Figure 2

Two subpictures of the generating pictures.

As it is seen in Figure 2(b), we also have the subpicture A R + , y Open image in new window (and similarly A R , y Open image in new window) with
exp T y x ( A R + , y ) = 1 + i + i 2 + + i μ 1 = i μ 1 i 1 Open image in new window
and
exp T y x ( A R , y ) = 1 + i + i 2 + + i λ 1 = i λ 1 i 1 . Open image in new window
By equality (6), we must have [ y i μ ] = [ y i λ ] Open image in new window. Hence, by [10], the subpicture C y , θ R Open image in new window with
ι ( C y , θ R ) = y i μ , τ ( C y , θ R ) = y i λ and exp S ( C y , θ R ) = i μ i λ k l Open image in new window
can be depicted as in Figure 3.
Figure 3

Subpicture C y , θ R Open image in new window of the generating picture.

After all, the whole generating pictures P S , x Open image in new window and P R , y Open image in new window can be drawn as in Figure 4.
Figure 4

Collection of the generating pictures of P M Open image in new window in ( 7 ).

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

Proposition 2.1 ([18])

Let p be a prime. Suppose that K θ A Open image in new window is a monoid with the associated monoid presentation P M Open image in new window, as in (7). Then P M Open image in new window is p-Cockcroft (equivalently efficient) if and only if
p k l , p i 1 , p | i μ i λ k l , p | i μ i λ i 1 . Open image in new window

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

Considering Theorem 1.1, one can say that the monoid presentation P M Open image in new window, as in (7), is efficient if and only if there is a prime p such that

In particular, if we choose exp S ( B S , x ) = i = 0 Open image in new window or 2, then P M Open image in new window will be inefficient.

Recall that, by the meaning of finite cyclic monoids, exp y ( S ) = k l Open image in new window cannot be equal to 0. We also note that a similar proof for the following result about minimal but inefficiency of P M Open image in new window can be found in [18].

Proposition 2.3 Let M be the semi-direct product of K by A, and let P M Open image in new window, as in (7), be the presentation for M where l , k , λ , μ , i Z + Open image in new window and l < k Open image in new window, λ < μ Open image in new window. If i = 2 Open image in new window and the subtraction k l Open image in new window is not even and not equal to 1, then P M Open image in new window is minimal but inefficient.

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

2.2 Part II: generating functions

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

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

Our first result of this section is related to the connection of the monoid presentation in (7) with array polynomials. In fact array polynomials S a n ( x ) Open image in new window are defined by means of the generating function
( e t 1 ) a e t x x ! = n = 0 S a n ( x ) t n n ! Open image in new window
(cf. [19, 20, 21]). According to the same references, array polynomials can also be defined as the form
S a n ( x ) = 1 a ! j = 0 a ( 1 ) a j ( a j ) ( x + j ) n . Open image in new window
(8)

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

From (5), we know that μ = λ + r Open image in new window, where 1 r μ 1 Open image in new window. Hence, by considering the meaning and conditions of Proposition 2.1, we obtain the following theorem as one of the main results of this paper.

Theorem 2.5 The efficient presentation P M Open image in new window defined in (7) has a set of generating functions
p 1 ( x ) = S n n ( x ) i S 0 1 ( x ) , p 2 ( y ) = ( k l ) S n n ( y ) , p 3 ( x ) = i λ ( i r 1 ) i 1 S n n ( x ) , p 4 ( y ) = i λ ( i r 1 ) k l S n n ( y ) , } Open image in new window
(9)

where S a n ( x ) Open image in new window and S a n ( y ) Open image in new window are defined as in (8).

Proof Let us consider the generating pictures P S , x Open image in new window, P R , y Open image in new window (in Figure 4) with their non-spherical subpictures defined in Figures 2 and 3, and the generating pictures of finite monogenic monoids defined in Figure 1. Recall that by counting the exponent sums of the discs R, S and T y x Open image in new window in the related pictures, the conditions of Proposition 2.1 have been obtained [18]. (For more similar results and applications, one can see the papers [10, 11].)

To reach our aim in the proof, we first need to calculate eval ( l ) ( P S , x ) Open image in new window, eval ( l ) ( P R , y ) Open image in new window, eval ( l ) ( P k , l m ) Open image in new window ( 1 m k 1 Open image in new window) and eval ( l ) ( P λ + r , λ n ) Open image in new window ( 1 n ( λ + r ) 1 Open image in new window). By Equations (1) and (2), we have
where y Open image in new window denotes the Fox derivation [23]. Also, for each 1 m k 1 Open image in new window and 1 n ( λ + r ) 1 Open image in new window,
eval ( l ) ( P k , l m ) = ( 1 y k m ) e S and eval ( l ) ( P λ + r , λ n ) = ( 1 x ( λ + r ) n ) e R . Open image in new window
Therefore, by the definition, the second Fox ideal I 2 ( l ) ( P M ) Open image in new window of the presentation P M Open image in new window in (7) is generated by the polynomial elements
1 x ( eval ( l ) ( B S , x ) ) , M S y , eval ( l ) ( A R + , x ) eval ( l ) ( A R , x ) , eval ( l ) ( C y , θ R ) , 1 y k 1 , 1 y k 2 , , 1 y , 1 x ( λ + r ) 1 , 1 x ( λ + r ) 2 , , 1 x . } Open image in new window
(10)
We need to keep our calculations going to other evaluations in the above polynomial elements. To do that, one can consider the augmentation map aug : Z M Z Open image in new window, b 1 Open image in new window. Under this map, it is easy to see that
aug ( eval ( l ) ( B S , x ) ) = exp S ( B S , x ) = i , aug ( M S y ) = exp y ( S ) = k l , aug ( eval ( l ) ( A R + , y ) eval ( l ) ( A R , y ) ) = exp T y x ( P R , y ) = i λ + r i λ i 1 , aug ( eval ( l ) ( C y , θ R ) ) = exp S ( P R , y ) = i λ + r i λ k l } Open image in new window
(11)
and for each 1 m k 1 Open image in new window and 1 n ( λ + r ) 1 Open image in new window,
aug ( eval ( l ) ( P k , l m ) ) = 0 and aug ( eval ( l ) ( P λ + r , λ n ) ) = 0 . Open image in new window
Now, by using (8) and keeping in our mind the coefficients of array polynomials are integer, we clearly have
S a n ( b ) = { b n ; a = 0 , b ; a = 0  and  n = 1 , 1 ; k = n  or  n = a = 0 . Open image in new window

Then, by reformulating the elements in (10) and (11) of the second Fox ideal I 2 ( l ) ( P M ) Open image in new window, we arrive at the functions in (9) as desired. □

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

Corollary 2.6 For any prime p, the presentation
P M = [ y , x ; y ( p + 1 ) [ ( p + 1 ) p 1 p ] + 1 = y , x p + 1 = x , y x = x y p + 1 ] Open image in new window
has a set of generating functions

In Proposition 2.3, the minimality (while satisfying inefficiency) of the presentation P M Open image in new window was expressed in (7). Thus, by considering the meaning and conditions of Proposition 2.3, we obtain the following theorem as another main result of this paper. Since the proof is quite similar to the proof of Theorem 2.5, we omit it.

Theorem 2.7 The inefficient but minimal presentation P M Open image in new window defined in (7) has a set of generating functions

where k l Open image in new window is an odd integer and S a n ( x ) Open image in new window and S a n ( y ) Open image in new window are defined as in (8).

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

Corollary 2.8 For an odd positive integer t, the presentation
P M = [ y , x ; y 4 t = y t , x 3 t = x t , y x = x y 2 ] Open image in new window
has a set of generating functions
Corollary 2.9 For any positive integers s and t with the condition s < t Open image in new window, the presentation
P M = [ y , x ; y 2 t + 1 = y 2 s , x 2 ( t s ) + 1 = x , y x = x y 2 ] Open image in new window
has a set of generating functions

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

At this point, we should note that for t 1 t 2 R + Open image in new window, γ C Open image in new window, a N 0 Open image in new window, generalized array type polynomials S a n ( x ; t 1 , t 2 ; γ ) Open image in new window related to the non-negative real parameters have been recently developed (in [20]) and some elementary properties including recurrence relations of these polynomials have been derived. In fact, by setting t 1 = γ = 1 Open image in new window and t 2 = e Open image in new window, 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 S a n ( x ; t 1 , t 2 ; γ ) Open image in new window.

The remaining goal of this section is to make a connection between the presentation P M Open image in new window in (7) and Stirling numbers of the second kind (cf. [20, 24, 25, 26, 27, 28] and the references of these papers). In fact, Stirling numbers of the second kind S ( n , a ) Open image in new window are defined by means of the generating function
( e t 1 ) a a ! = n = 0 S ( n , a ) t n n ! Open image in new window
(see [27, 28]). According to [[20], Theorem 1, Remark 2], Stirling numbers can also be defined as the form
S ( n , a ) = 1 a ! j = 0 a ( 1 ) j ( a j ) ( k j ) n . Open image in new window
We remind that these numbers satisfy the well-known properties
S ( n , a ) = { 1 ; a = 1  or  a = n , ( n 2 ) ; a = n 1 , δ n , 0 ; a = 0 , Open image in new window

where δ n , 0 Open image in new window denotes the Kronecker symbol (see [27, 28]). It is known that Stirling numbers are used in combinatorics, in number theory, in discrete probability distributions for finding higher-order moments, etc. We finally note that since S ( n , a ) Open image in new window is the number of ways to partition a set of n objects into k groups, these numbers find an application area in combinatorics and in theory of partitions.

In addition to the above formulas, for S ( n , a ) Open image in new window, by [20, 26, 27], we also have
x n = a = 0 n ( x a ) a ! S ( n , a ) Open image in new window
(12)
as a formula for Stirling numbers. Therefore, by taking n = 1 Open image in new window and n = 0 Open image in new window in Equation (12), the polynomial elements of the second Fox ideal I 2 ( l ) ( P M ) Open image in new window of the presentation P M Open image in new window in (7) can be restated as follows:
x 0 i x 1 = a = 0 0 ( x a ) a ! S ( 0 , a ) i a = 0 1 ( x a ) a ! S ( 1 , a ) , ( k l ) y 0 = ( k l ) a = 0 0 ( y a ) a ! S ( 0 , a ) , i λ ( i r 1 ) i 1 x 0 = i λ ( i r 1 ) i 1 a = 0 0 ( x a ) a ! S ( 0 , a ) , i λ ( i r 1 ) k l y 0 = i λ ( i r 1 ) k l a = 0 0 ( y a ) a ! S ( 0 , a ) . } Open image in new window
(13)

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 P M Open image in new window in (7) has a set of generating functions in terms of Stirling numbers as given in (13). By taking i = 2 Open image in new window and k l Open image in new window is an odd positive integer, we get a set of generating functions in terms of Stirling numbers for the inefficient but minimal presentation of the form as defined in (7).

Furthermore, in a recent work, Simsek [20] has constructed the generalized γ-Stirling numbers of the second kind S ( n , v ; a , b ; γ ) Open image in new window related to non-negative real parameters ( a , b R + Open image in new window, a b Open image in new window, a complex number γ and v N 0 Open image in new window). In fact, this new generalization is defined by the generating function as the equality
f S , v ( t ; a , b ; γ ) = ( γ b t a t ) v v ! = n = 0 S ( n , v ; a , b ; γ ) t n n ! . Open image in new window
(14)
By setting a = 1 Open image in new window and b = e Open image in new window in (14), one can obtain the γ-Stirling numbers of the second kind S ( n , v ; γ ) Open image in new window which are defined by the generating function
( γ e t 1 ) v v ! = n = 0 S ( n , v ; γ ) t n n ! Open image in new window

(see [27, 28]). According to the same references, by substituting γ = 1 Open image in new window into the above equation, the Stirling numbers of the second kind S ( n , v ) Open image in new window are obtained.

By considering this new generalization S ( n , v ; a , b ; γ ) Open image in new window, in [[20], Theorem 1], the equality
S ( n , v ; a , b ; γ ) = 1 v ! j = 0 n ( 1 ) j ( v j ) γ v j ( j ln a + ( v j ) ln b ) n Open image in new window
(15)
has also been obtained for γ-Stirling numbers of the second kind. In fact, by setting a = 1 Open image in new window and b = e Open image in new window in (15), one can get the following equality on γ-Stirling numbers:
S ( n , v ; γ ) = 1 v ! j = 0 v ( v j ) λ ( v j ) ( 1 ) j ( v j ) n Open image in new window
(16)

(see [27, 28]).

Hence, we can present the following note.

Remark 2.13 It is clearly seen that Stirling numbers have been only considered in Theorems 2.5 and 2.7 (and the corollaries about them). However, one can also study the γ-Stirling numbers S ( n , v ; γ ) Open image in new window defined in (16) and generalized γ-Stirling numbers S ( n , v ; a , b ; γ ) Open image in new window defined in (15) to obtain different types of generating functions.

3 The constant function related to main results

In Theorems 2.5, 2.7 and 2.12, we have actually used
i λ ( i r 1 ) i 1 and i λ ( i r 1 ) k l Open image in new window

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

Hence, let us consider the analytic function
f ( z , r , λ , k , l ) = z λ + r z λ k l , Open image in new window
(17)
where z C Open image in new window and r , λ , k , l Z + Open image in new window. To reach our aim, let us first replace the complex element z by a positive integer i in (17), and then apply some fundamental algebraic progress to it. Therefore,
f ( i , r , λ , k , l ) = i λ + r i λ k l = i λ k l ( i r 1 ) = i λ k l ( i 1 ) ( i r 1 + + 1 ) = i λ k l ( i 1 ) ϕ r 1 ( i ) , Open image in new window
(18)
where ϕ r 1 ( i ) Open image in new window is a cyclotomic polynomial having degree r 1 Open image in new window. By considering finite powers of the function f ( z , r , λ , k , l ) Open image in new window given in (17), we can get
Y ( z ) = [ f ( z , r , λ , k , l ) ] m = m ! m ! ( z λ + r z λ k l ) m , Open image in new window
(19)

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

Now, if we replace z by e t Open image in new window, then we get
Y ( e t ) = m ! m ! ( e t ( λ + r ) e t λ k l ) m = m ! k l e m t λ ( e t r 1 ) m m ! = m ! k l e m t λ n = 0 S ( n , m ) r n t n n ! , Open image in new window
(20)
where  S ( n , m )  defines the Stirling numbers of the second-kind = m ! k l n = 0 m n λ n t n n ! n = 0 S ( n , m ) r n t n n ! . Open image in new window
(21)
Further, by applying the Cauchy multiplication in (21), we finally obtain
Y ( e t ) = m ! k l n = 0 [ a = 0 n ( n a ) S ( a , m ) r a m n a λ n a ] t n n ! . Open image in new window

All these above processes imply the following result.

Theorem 3.1
( f ( e t , r , λ , k , l ) ) m = n = 0 a n k l t n n ! , Open image in new window
(22)
where
a n = m ! a = 0 n ( n a ) S ( a , m ) r a m n a λ n a Open image in new window

and S ( a , m ) Open image in new window denotes the Stirling numbers of second kind.

Some properties of the function Y ( z ) = [ f ( z , r , λ , k , l ) ] m Open image in new window in (19) can be expressed as follows:

Remark 3.2 Setting m = 1 Open image in new window in (22), one can easily see that
f ( e t , r , λ , k , l ) = n = 0 [ a = 0 n ( n a ) r a λ n a k l ] t n n ! , Open image in new window

since S ( n , 1 ) = 1 Open image in new window.

By considering [[20], Eq. (3.2)] and Equation (20), we can extend Remark 3.2 to a general natural number m > 1 Open image in new window as in the following theorem.

Theorem 3.3
( e t ( λ + r ) e t λ k l ) m = m ! k l n = 0 S m n ( m λ ) r n t n n ! , Open image in new window

where S m n ( m λ ) Open image in new window 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 C Open image in new window and z R Open image in new window, analytic and functional equations can be studied.

  • As we have partially done in the above, replacing z by e t Open image in new window, one can study the generating functions of array polynomials and Stirling numbers.

3.1 Some other properties over this constant

Let us consider the first derivation of the function in (17). We then have
f ( z , r , λ , k , l ) = ( λ + r ) z λ + r 1 λ z λ 1 k l = λ z λ + r 1 λ z λ 1 k l + r z λ + r 1 k l = λ z λ 1 k l ( z r 1 ) + r z λ + r 1 k l , Open image in new window
or equivalently,
f ( z , r , λ , k , l ) = λ z λ 1 k l ( z 1 ) ϕ r 1 ( z ) + r z λ + r 1 k l . Open image in new window
(23)
In (23), replacing z by i, we get
f ( i , r , λ , k , l ) = λ i λ 1 k l ( i 1 ) ϕ r 1 ( i ) + r i λ + r 1 k l = λ [ i λ 1 k l ( i 1 ) ϕ r 1 ( i ) ] + r i λ + r 1 k l , Open image in new window
and then by using (18), we have
f ( i , r , λ , k , l ) = λ i f ( i , r , λ , k , l ) + r i λ + r 1 k l . Open image in new window
(24)
As the next step, let us calculate the second derivative of f ( z , r , λ , k , l ) Open image in new window:
f ( z , r , λ , k , l ) = λ ( λ + r 1 ) z λ + r 2 λ ( λ 1 ) z λ 2 k l + r ( λ + r 1 ) z λ + r 2 k l = λ ( λ 1 ) z λ 2 k l ( z r 1 ) + r z λ + r 2 k l + r ( λ + r 1 ) z λ + r 2 k l = λ ( λ 1 ) z λ 2 k l ( z r 1 ) + r z λ + r 2 k l ( 2 λ + r 1 ) , Open image in new window
and by collecting some terms in brackets, we get
f ( z , r , λ , k , l ) = ( λ 1 ) z ( λ z λ 1 k l ( z 1 ) ϕ r 1 ( z ) + r z λ + r 1 k l ) + r z λ + r 2 k l ( λ + r ) . Open image in new window
Now, using (24), the second derivative of the function f ( z , r , λ , k , l ) Open image in new window will be equal to
f ( z , r , λ , k , l ) = ( λ 1 ) z f ( z , r , λ , k , l ) + r z λ + r 2 k l ( λ + r ) . Open image in new window
(25)
Replacing z by i in (25) and using (18), we obtain
f ( i , r , λ , k , l ) = ( λ 1 ) i f ( i , r , λ , k , l ) r i λ + r 1 + r i λ + r 2 k l ( λ + r ) = λ ( λ 1 ) i 2 f ( i , r , λ , k , l ) + r ( λ 1 ) i i λ + r 1 k l + r i λ + r 2 k l ( λ + r ) = λ ( λ 1 ) i 2 f ( i , r , λ , k , l ) + [ r ( λ 1 ) i i λ + r 1 k l + r ( 2 λ + r 1 ) k l ] i λ + r 2 . Open image in new window
By iterating these above derivations for the variable z, and then replacing z by i, we finally obtain
f ( m ) ( i , r , λ , k , l ) = λ ( λ 1 ) ( λ m + 1 ) i m f ( i , r , λ , k , l ) + A i λ + r m = m ! ( λ m ) i m f ( i , r , λ , k , l ) + A i λ + r m , Open image in new window

where A Open image in new window stands for some constants.

This above theory is related to the functional equations. In fact, these above progresses show that the presentation P M Open image in new window 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

13663_2012_353_MOESM1_ESM.eps (176 kb)
Authors’ original file for figure 1
13663_2012_353_MOESM2_ESM.eps (153 kb)
Authors’ original file for figure 2
13663_2012_353_MOESM3_ESM.eps (141 kb)
Authors’ original file for figure 3
13663_2012_353_MOESM4_ESM.eps (196 kb)
Authors’ original file for figure 4

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Copyright information

© Çevik et al.; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  • A Sinan Çevik
    • 1
    Email author
  • I Naci Cangül
    • 2
  • Yılmaz Şimşek
    • 3
  1. 1.Department of Mathematics, Faculty of ScienceSelçuk UniversityKonyaTurkey
  2. 2.Department of Mathematics, Faculty of Arts and ScienceUludag UniversityBursaTurkey
  3. 3.Department of Mathematics, Faculty of Art and ScienceAkdeniz UniversityAntalyaTurkey

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