This main section will be given as three subsections under the names of Generating pictures (as Part I), Characteristic polynomials (as Part II) and Array polynomials (as Part III). In our study, we will consider a special presentation {\mathcal{P}}_{E} (see (2) below). Since we will define characteristic polynomials and generating functions by considering the exponent sums of the generating pictures over {\mathcal{P}}_{E}, the first subsection is aimed at presenting these generating pictures and the related results about them.
2.1 Part I: Generating pictures of {\mathcal{P}}_{E}
It is strictly referred to [6–8] for fundamentals and properties of the algebraic subject used in this subsection. We further note that most of the material in here can also be found in [9].
We will mainly present the efficiency (equivalently, pCockcroft property for a prime p) for the semidirect product of free abelian monoid {K}_{2} having rank two by a finite cyclic monoid A of order k. Hence, for 1\le l<k and l,k\in {\mathbb{Z}}^{+}, let {\mathcal{P}}_{A}=[x;{x}^{k}={x}^{l}] be a presentation of A and {\mathcal{P}}_{{K}_{2}}=[{y}_{1},{y}_{2};{y}_{1}{y}_{2}={y}_{2}{y}_{1}] be a presentation of {K}_{2}. Suppose that ψ is the endomorphism {\psi}_{\mathcal{M}} of {K}_{2}, where \mathcal{M}=\left[\begin{array}{cc}{\alpha}_{11}& {\alpha}_{12}\\ {\alpha}_{21}& {\alpha}_{22}\end{array}\right] such that the entries {\alpha}_{ij}’s are the positive integers defined by {y}_{1}\mapsto {{y}_{1}}^{{\alpha}_{11}}{{y}_{2}}^{{\alpha}_{12}} and {y}_{2}\mapsto {{y}_{1}}^{{\alpha}_{21}}{{y}_{2}}^{{\alpha}_{22}}. Hence the mapping x\mapsto {\psi}_{x} (x\in \mathbf{x}) induces a welldefined monoid homomorphism \theta :A\u27f6End({K}_{2}) if and only if {\mathcal{M}}_{[{x}^{k}]}={\mathcal{M}}_{[{x}^{l}]}, or equivalently,
{\mathcal{M}}^{k}\equiv {\mathcal{M}}^{l}modd,
(1)
where d\mid (kl) (see [9] for the details). Since there exists an ordering between the relators of A, that is, for k,l\in {\mathbb{Z}}^{+}, we have 1\le l<k, it implies that to define an induced homomorphism \theta :A\u27f6End({K}_{2}), that is, to define {K}_{2}{\u22ca}_{\theta}A, we must take congruence relation between {\mathcal{M}}^{k} and {\mathcal{M}}^{l} as given in (1) with the assumption d\mid (kl). By [9], the k th and l th powers of the matrices can be obtained by ordinary matrix multiplication. Thus, let us suppose that the k th (k\in {\mathbb{Z}}^{+}) power of ℳ is defined as
\begin{array}{rcl}{\mathcal{M}}^{k}& =& \left[\begin{array}{cc}{a}_{k1}& {b}_{k1}\\ {c}_{k1}& {z}_{k1}\end{array}\right]\left[\begin{array}{cc}{\alpha}_{11}& {\alpha}_{12}\\ {\alpha}_{21}& {\alpha}_{22}\end{array}\right]\\ =& \left[\begin{array}{cc}{a}_{k1}{\alpha}_{11}+{b}_{k1}{\alpha}_{21}& {a}_{k1}{\alpha}_{12}+{b}_{k1}{\alpha}_{22}\\ {c}_{k1}{\alpha}_{11}+{z}_{k1}{\alpha}_{21}& {c}_{k1}{\alpha}_{12}+{z}_{k1}{\alpha}_{22}\end{array}\right]=\left[\begin{array}{cc}{a}_{k}& {b}_{k}\\ {c}_{k}& {z}_{k}\end{array}\right],\end{array}
while, applying a similar idea, the l th (l\in {\mathbb{Z}}^{+}) power of ℳ is defined as {\mathcal{M}}^{l}=\left[\begin{array}{cc}{a}_{l}& {b}_{l}\\ {c}_{l}& {z}_{l}\end{array}\right]. Now suppose that (1) holds. Then the semidirect product E={K}_{2}{\u22ca}_{\theta}A has a presentation
{\mathcal{P}}_{E}=[{y}_{1},{y}_{2},x;S,R,{T}_{{y}_{1}x},{T}_{{y}_{2}x}],
(2)
where S:{y}_{1}{y}_{2}={y}_{2}{y}_{1}, R:{x}^{k}={x}^{l}, {T}_{{y}_{1}x}:{y}_{1}x=x{y}_{1}^{{\alpha}_{11}}{y}_{2}^{{\alpha}_{12}} and {T}_{{y}_{2}x}:{y}_{2}x=x{y}_{1}^{{\alpha}_{21}}{y}_{2}^{{\alpha}_{22}}, respectively.
In the rest of this paper, we will assume that Equality (1) always holds when we talk about the semidirect product E of {K}_{2} by A.
We know that the trivializer set (see [7]) of {\mathbf{X}}_{\mathbf{E}} of \mathcal{D}({\mathcal{P}}_{E}) consists of the trivializer set {\mathbf{X}}_{{\mathbf{K}}_{2}} of \mathcal{D}({\mathcal{P}}_{{K}_{2}}), {\mathbf{X}}_{\mathbf{A}} of \mathcal{D}({\mathcal{P}}_{A}) and the sets {\mathbf{C}}_{1}, {\mathbf{C}}_{2} (see [[8], Lemma 1.5]). In our case, by [16], {\mathbf{X}}_{{\mathbf{K}}_{2}} is equal to the empty set since, for the relator S, we have \iota ({S}_{+})\ne \iota ({S}_{}). Thus {\mathcal{P}}_{{K}_{2}} is aspherical and so pCockcroft for any prime p. Nevertheless, the trivializer set {\mathbf{X}}_{\mathbf{A}} of the Squier complex \mathcal{D}({\mathcal{P}}_{A}) can be found in [[7], Lemma 4.4]. Finally, the subsets {\mathbf{C}}_{1} and {\mathbf{C}}_{2} contain the generating pictures {\mathbb{P}}_{S,x} (which contains a nonspherical subpicture {\mathbb{B}}_{S,x} as depicted in [7]), {\mathbb{P}}_{R,{y}_{1}} and {\mathbb{P}}_{R,{y}_{2}} of the trivializer set {\mathbf{X}}_{\mathbf{E}}. These pictures can be presented as in Figure 1(a) and (b).
For simplicity, let us define the sum of each entries of power matrices
\begin{array}{l}{a}_{0}+{a}_{1}+\cdots +{a}_{k1}\phantom{\rule{1em}{0ex}}\text{as}{\mathbf{a}}_{k},\phantom{\rule{2em}{0ex}}{a}_{0}+{a}_{1}+\cdots +{a}_{l1}\phantom{\rule{1em}{0ex}}\text{as}{\mathbf{a}}_{l},\\ {b}_{0}+{b}_{1}+\cdots +{b}_{k1}\phantom{\rule{1em}{0ex}}\text{as}{\mathbf{b}}_{k},\phantom{\rule{2em}{0ex}}{b}_{0}+{b}_{1}+\cdots +{b}_{l1}\phantom{\rule{1em}{0ex}}\text{as}{\mathbf{b}}_{l},\\ {c}_{0}+{c}_{1}+\cdots +{c}_{k1}\phantom{\rule{1em}{0ex}}\text{as}{\mathbf{c}}_{k},\phantom{\rule{2em}{0ex}}{c}_{0}+{c}_{1}+\cdots +{c}_{l1}\phantom{\rule{1em}{0ex}}\text{as}{\mathbf{c}}_{l},\\ {z}_{0}+{z}_{1}+\cdots +{z}_{k1}\phantom{\rule{1em}{0ex}}\text{as}{\mathbf{z}}_{k},\phantom{\rule{2em}{0ex}}{z}_{0}+{z}_{1}+\cdots +{z}_{l1}\phantom{\rule{1em}{0ex}}\text{as}{\mathbf{z}}_{l}.\end{array}\}
(3)
Suppose that the positive integer d, defined in (1), is equal to a prime p such that p\mid (kl). Then, in [9], the following result has been recently obtained.
Proposition 1 ([9])
Let p be a prime or 0. Then the presentation {\mathcal{P}}_{E}, as in (2), for the monoid E={K}_{2}{\u22ca}_{\theta}A is pCockcroft if and only if

(a)
det\mathcal{M}\equiv 1modp,

(b)
\begin{array}{c}{\mathbf{a}}_{k}\equiv {\mathbf{z}}_{l}modp,\phantom{\rule{2em}{0ex}}{\mathbf{b}}_{k}\equiv {\mathbf{c}}_{l}modp,\hfill \\ {\mathbf{c}}_{k}\equiv {\mathbf{b}}_{l}modp,\phantom{\rule{2em}{0ex}}{\mathbf{z}}_{k}\equiv {\mathbf{a}}_{l}modp.\hfill \end{array}
According to the above proposition, let us take an efficient presentation
{\mathcal{P}}_{E}=[{y}_{1},{y}_{2},x;{y}_{1}{y}_{2}={y}_{2}{y}_{1},{x}^{2p+1}=x,{y}_{1}x=x{y}_{1}^{{\alpha}_{11}}{y}_{2}^{{\alpha}_{12}},{y}_{2}x=x{y}_{1}^{{\alpha}_{21}}{y}_{2}^{{\alpha}_{22}}].
(4)
Suppose that p is an odd prime. Then, in particular, {\mathcal{P}}_{E} in (4) is not efficient if det\mathcal{M} is either equivalent to 0 or p1 by modulo p. Therefore one of the consequences of Proposition 1 is the following.
Proposition 2 ([9])
The presentation {\mathcal{P}}_{E} in (4) is minimal but inefficient if p is an odd prime and
\mathit{\text{either}}\phantom{\rule{1em}{0ex}}\{\begin{array}{l}{\alpha}_{11}=p1,\\ {\alpha}_{12}={\alpha}_{21}=0,\\ {\alpha}_{22}=1\end{array}\phantom{\rule{1em}{0ex}}\mathit{\text{or}}\phantom{\rule{1em}{0ex}}\{\begin{array}{l}{\alpha}_{11}=1,\\ {\alpha}_{12}={\alpha}_{21}=0,\\ {\alpha}_{22}=p1.\end{array}
2.2 Part II: Characteristic polynomials over {\mathcal{P}}_{E}
Let us reconsider the semidirect product of {K}_{2} by A with a presentation {\mathcal{P}}_{E} as defined in (2). Now it is well known that if one wants to define such a presentation {\mathcal{P}}_{E}, then it must satisfy condition (1). Therefore we certainly have 2\times 2matrices ℳ, {\mathcal{M}}^{k} and {\mathcal{M}}^{l}, and so we can consider the related characteristic polynomials of these matrices.
Let {\lambda}_{1} and {\lambda}_{2} be the only eigenvalues of the matrix ℳ. Since the entries of ℳ are positive integers, {\lambda}_{1} and {\lambda}_{2} could be any numbers including complex ones. As a restriction, throughout all paper, we will assume that these eigenvalues are real. By using the basic fact of linear algebra, we then have the eigenvalues of {\mathcal{M}}^{k} as {\lambda}_{1}^{k} and {\lambda}_{2}^{k} while the eigenvalues of {\mathcal{M}}^{l} as {\lambda}_{1}^{l} and {\lambda}_{2}^{l}. Thus, for a variable ν, the characteristic polynomials over each of the matrices ℳ, {\mathcal{M}}^{k} and {\mathcal{M}}^{l} will be of the form
\begin{array}{l}{p}_{k}^{l}{(\nu )}_{\lambda}={\nu}^{2}({\lambda}_{1}+{\lambda}_{2})\nu +{\lambda}_{1}{\lambda}_{2}\\ {p}_{k}^{l}{(\nu )}_{{\lambda}^{k}}={\nu}^{2}({\lambda}_{1}^{k}+{\lambda}_{2}^{k})\nu +{\lambda}_{1}^{k}{\lambda}_{2}^{k}\\ {p}_{k}^{l}{(\nu )}_{{\lambda}^{l}}={\nu}^{2}({\lambda}_{1}^{l}+{\lambda}_{2}^{l})\nu +{\lambda}_{1}^{l}{\lambda}_{2}^{l}\end{array}\},
(5)
respectively.
Now let us think of (5) as a piece of the system of characteristic polynomials that are obtained from any 2\times 2matrix. In fact, there are an infinite number of polynomials of the type (5) since one can find an infinite number of matrices having positive integer entries that satisfy condition (1). On the other hand, by the definition of finite monogenic monoids and semigroups (see [17]), for a fixed value of k, one can choose the value l from the set \{1,2,\dots ,k1\}. It is clear that each of these systems in (5) will be constructed as a choice of l in this set, and so we will have k1 times different systems of characteristic polynomials as in (5) such that each of them contains an infinite number of polynomials. The following proposition will be based on this fact.
Proposition 3 Each characteristic polynomial obtained from 2\times 2 matrices, as in system (5), appears to be a congruence class.
Proof Before giving the proof, we note that this result can of course be adapted to n\times n matrices, which will not be needed in here.
The normal form theorem [18], NFT for short, basically says that each congruence class contains a unique reduced word. It is well known in the branch of combinatorial group theory that the idea of this theorem is the main point when one tries to define a presentation for the related algebraic structure. Therefore, since each presentation {\mathcal{P}}_{A} and {\mathcal{P}}_{K} has been obtained by considering NFT, the presentation {\mathcal{P}}_{E} in (2) must also satisfy NFT. (Although we reminded this theorem by considering only the words over free generators, the idea of NFT can actually be seen in ℚ as fractals or in matrix theory as echelon forms.)
For a fixed modulo d, let us think about the set, say {\mathrm{\Delta}}_{d}^{\nu}, of all characteristic polynomials, written as in system (5), having the condition {\mathcal{M}}^{k}\equiv {\mathcal{M}}^{l}modd, where d\mid kl and l\in \{1,2,\dots ,k1\}. We note that the cardinality s({\mathrm{\Delta}}_{d}^{\nu})=k1.
Nevertheless, for each different 1\le l\le k1, since one can find matrices satisfying the condition in (1), the set {\mathrm{\Delta}}_{d}^{\nu} can be constructed as a union of k1 times congruence classes, i.e., lclasses. Moreover, again by (1), for each 1\le l\le k1, since we also have
{\lambda}_{i}^{k}\equiv {\lambda}_{i}^{l}\equiv {\lambda}_{i}modd,\phantom{\rule{1em}{0ex}}\text{where}1\le i\le 2,
(6)
each of these lclasses contains the characteristic polynomials having eigenvalues {\lambda}_{i}^{k}, {\lambda}_{i}^{l} and {\lambda}_{i} (for 1\le i\le 2). On the other hand, by (6), {p}_{k}^{l}{(\nu )}_{\lambda} can be taken as the simplest element (characteristic polynomial) having eigenvalues {\lambda}_{1} and {\lambda}_{2} in each lclass.
Thus, let us choose a power l from the set \{1,2,\dots ,k1\} for a fixed k. Then the set {\mathrm{\Delta}}_{d}^{\nu} will be constructed for a suitable modulo d according to our choice of l, which satisfies (1). Now, by (6), we have
{p}_{k}^{l}{(\nu )}_{\lambda}\equiv {p}_{k}^{l}{(\nu )}_{{\lambda}^{k}}\equiv {p}_{k}^{l}{(\nu )}_{{\lambda}^{l}}modd
in system (5). In other words, for a fixed k and modulo d, there exists \overline{{p}_{k}^{l}{(\nu )}_{\lambda}}=\overline{{p}_{k}^{l}{(\nu )}_{{\lambda}^{k}}}=\overline{{p}_{k}^{l}{(\nu )}_{{\lambda}^{l}}}, where each class contains an infinite number of elements (polynomials). Therefore
\overline{{p}_{k}^{l}{(\nu )}_{\lambda}}=\{{p}_{k}^{l}{(\nu )}_{\lambda},{p}_{k}^{l}{(\nu )}_{{\lambda}^{k}},{p}_{k}^{l}{(\nu )}_{{\lambda}^{l}},\dots \},
where 1\le l\le k1. As a summary, for a chosen d,
{\mathrm{\Delta}}_{d}^{\nu}=\{\overline{{p}_{k}^{l}{(\nu )}_{\lambda}}:k\in {\mathbb{Z}}^{+}\text{is fixed,}1\le l\le k1,d\text{is fixed with}d\mid kl\}
such that {\bigcup}_{1\le l\le k1}\overline{{p}_{k}^{l}{(\nu )}_{\lambda}}={\mathrm{\Delta}}_{d}^{\nu} and {\bigcap}_{1\le l\le k1}\overline{{p}_{k}^{l}{(\nu )}_{\lambda}}=\mathrm{\varnothing}.
Hence the result. □
As a next step of Proposition 3, the following result basically states that each semidirect product presentation as in (2) which is obtained by condition (1) has a characteristic polynomial.
Theorem 1 The presentation {\mathcal{P}}_{E} in (2) is represented by a unique (up to equivalence) characteristic polynomial defined as in system (5).
Proof By Proposition 3 and as a result of NFT, the simplest (unique reduced) characteristic polynomial in system (5) represents the congruence class of the related lclasses. From (1), we have {a}_{k}\equiv {a}_{l}modd, {b}_{k}\equiv {b}_{l}modd, {c}_{k}\equiv {c}_{l}modd and {z}_{k}\equiv {z}_{l}modd, where d\mid (kl). Thus, by (6), we clearly obtain the equivalences
{\lambda}_{1}^{k}+{\lambda}_{2}^{k}\equiv {\lambda}_{1}^{l}+{\lambda}_{2}^{l}\equiv {\lambda}_{1}+{\lambda}_{2}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\lambda}_{1}^{k}{\lambda}_{2}^{k}\equiv {\lambda}_{1}^{l}{\lambda}_{2}^{l}\equiv {\lambda}_{1}{\lambda}_{2}
(7)
by modulo d. Shortly, tr({\mathcal{M}}^{k})\equiv tr({\mathcal{M}}^{l})\equiv tr(\mathcal{M})modd, where tr(\cdot ) denotes the trace of these matrices, and det{\mathcal{M}}^{k}\equiv det{\mathcal{M}}^{l}\equiv det\mathcal{M}modd. This implies that each characteristic polynomial in system (5) is congruent to another by modulo d, and so the simplest polynomial
{\nu}^{2}tr(\mathcal{M})\nu +det\mathcal{M}
(8)
can be chosen as a representative characteristic polynomial for the presentation {\mathcal{P}}_{E} in (2) since all these polynomials are in the same lclass. As a result, since {\mathcal{P}}_{E} is obtained by using both unique reduced words (according to NFT) and the matrix ℳ as the endomorphism {\psi}_{\mathcal{M}}, polynomial (8) represents this presentation uniquely as a characteristic polynomial. □
As a consequence of Theorem 1 and Proposition 1, we can give the following result which is a simpler version of Proposition 1 since the efficiency conditions can be expressed as a unique statement.
Theorem 2 For any prime p or 0, the presentation {\mathcal{P}}_{E} in (2) is pCockcroft if and only if {\lambda}_{1}{\lambda}_{2}\equiv 1modp, where {\lambda}_{1}<{\lambda}_{2} are the eigenvalues of the 2\times 2matrix ℳ.
Proof Now, by Theorem 1, it is known that {\mathcal{P}}_{E} in (2) has a characteristic polynomial as in (8) with a base 2\times 2matrix ℳ. Let p be any prime or 0, and let {\lambda}_{1}<{\lambda}_{2} be the eigenvalues of ℳ. Thus, we clearly have tr(\mathcal{M})={\lambda}_{1}+{\lambda}_{2} and det(\mathcal{M})={\lambda}_{1}{\lambda}_{2}. By Proposition 1, since the sufficiency part is clear, let us show the necessity of the proof.
Assume that det(\mathcal{M})\equiv 1modp. Also, let us consider the power matrices {\mathcal{M}}^{k} and {\mathcal{M}}^{l} which certainly exist since {\mathcal{P}}_{E} in (2) is a semidirect product presentation. So, {\lambda}_{1}^{k}{\lambda}_{2}^{k}\equiv 1modp and {\lambda}_{1}^{l}{\lambda}_{2}^{l}\equiv 1modp. In other words, by considering the sums presented in (3),
{\mathbf{a}}_{k}{\mathbf{z}}_{k}{\mathbf{b}}_{k}{\mathbf{c}}_{k}\equiv 1modp\phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}{\mathbf{a}}_{k}{\mathbf{z}}_{k}=p{t}_{1}+1+{\mathbf{b}}_{k}{\mathbf{c}}_{k}
and
{\mathbf{a}}_{l}{\mathbf{z}}_{l}{\mathbf{b}}_{l}{\mathbf{c}}_{l}\equiv 1modp\phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}{\mathbf{a}}_{l}{\mathbf{z}}_{l}=p{t}_{2}+1+{\mathbf{b}}_{l}{\mathbf{c}}_{l},
for some integers {t}_{1} and {t}_{2}. Hence, by applying side by removing, we get
{\mathbf{a}}_{k}{\mathbf{z}}_{k}{\mathbf{b}}_{k}{\mathbf{c}}_{k}\equiv {\mathbf{a}}_{l}{\mathbf{z}}_{l}{\mathbf{b}}_{l}{\mathbf{c}}_{l}modp,
(9)
which actually implies the truthfulness of the second part of (7).
On the other hand, by Theorem 1,
{\mathbf{a}}_{k}{\mathbf{z}}_{k}\equiv {\mathbf{a}}_{l}{\mathbf{z}}_{l}modp
since each characteristic polynomial in system (5) must be in the same lclass. From the above equivalence, we definitely get {\mathbf{a}}_{k}\equiv {\mathbf{z}}_{l}modp and {\mathbf{a}}_{l}\equiv {\mathbf{z}}_{k}modp since our assumption det(\mathcal{M})\equiv 1modp does not permit another equivalence option. Furthermore, by keeping in our mind these last equivalences, if we consider (9) again, then we can get {\mathbf{b}}_{k}\equiv {\mathbf{c}}_{l}modp and {\mathbf{c}}_{k}\equiv {\mathbf{b}}_{l}modp since again the assumption on the determinant is enforced to not get another equivalence option.
Notice that the above processes show that this unique assumption on the determinant of ℳ implies the remaining conditions of Proposition 1. Thus, as a next step of Proposition 1, we clearly get {\mathcal{P}}_{E} is pCockcroft (equivalently, efficient) for any prime p or 0. □
Example 1 By considering the matrix ℳ, one can give the following examples for Theorem 2:

(i)
For an odd prime p, if we take {\alpha}_{11}=1, {\alpha}_{12}=t (t\in {\mathbb{Z}}^{+}), {\alpha}_{21}=0 and {\alpha}_{22}=1, then we get {\lambda}_{1}{\lambda}_{2}=1 while {\mathcal{M}}^{p+1}\equiv {\mathcal{M}}^{1}modp. Hence, we get an efficient presentation. However, for even prime, we can still get a semidirect product presentation for the same matrix entries since {\mathcal{M}}^{3}\equiv {\mathcal{M}}^{1}mod2. But the presentation will be inefficient.

(ii)
For any prime p and t\in {\mathbb{Z}}^{+}, again by taking
\left[\begin{array}{cc}1& t\\ 0& 1\end{array}\right]\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}\left[\begin{array}{cc}1& 0\\ t& 1\end{array}\right],
we can get {\mathcal{M}}^{2p+1}\equiv {\mathcal{M}}^{1}modp, and also {\lambda}_{1}{\lambda}_{2}=1. Hence we obtain an efficient presentation which can always be represented by a characteristic polynomial {(\nu 1)}^{2} for both cases.

(iii)
Similarly, for any prime p, by choosing either {\alpha}_{11}=pt+1 (t\in {\mathbb{Z}}^{+}), {\alpha}_{12}=0, {\alpha}_{21}=0 and {\alpha}_{22}=1 or {\alpha}_{11}=1, {\alpha}_{21}=0, {\alpha}_{12}=0 and {\alpha}_{22}=pt+1 (t\in {\mathbb{Z}}^{+}), we can still obtain efficient presentations with the equivalence {\mathcal{M}}^{2p+1}\equiv {\mathcal{M}}^{1}modp. Further, the representative characteristic polynomial for the related presentation is of the form {\nu}^{2}(pt+2)\nu +pt+1.

(iv)
The above examples can be extended for {\mathcal{M}}^{np+1}\equiv {\mathcal{M}}^{1}modp, where n\in {\mathbb{Z}}^{+}.
As another consequence of Theorem 1, we obtain the following result by considering Proposition 2.
Theorem 3 For an odd prime p, the presentation {\mathcal{P}}_{E} in (4) is minimal but inefficient if ℳ is a diagonal matrix and the representative characteristic polynomial is of the form {\nu}^{2}p\nu +(p1) which has a strict ordering 1<(p1) between roots.
Proof Let ℳ be a diagonal matrix. For an odd prime p, let {\nu}^{2}p\nu +(p1) be the representative characteristic polynomial for {\mathcal{P}}_{E}. Clearly, the roots 1 and p1 will be the eigenvalues {\lambda}_{1} and {\lambda}_{2} of ℳ. Then, by (8), we have tr(\mathcal{M})=p and det(\mathcal{M})=p1. By Proposition 2, this determinant value implies that {\mathcal{P}}_{E} is inefficient. Notice that since {\mathcal{P}}_{E} in (4) is a semidirect product presentation, {\mathcal{M}}^{2p+1}\equiv \mathcal{M}modp must be held. Therefore we have {\lambda}_{1}^{2p+1}\equiv {\lambda}_{1}modp. So, by Proposition 3, the polynomial with roots {\lambda}_{1}^{2p+1} and {\lambda}_{2} is in the same congruence class as the simplest characteristic polynomial {\nu}^{2}p\nu +(p1). In addition, by the assumption, the entries of ℳ should be either {\alpha}_{11}=1, {\alpha}_{22}=p1 or vice versa {\alpha}_{12}=0={\alpha}_{21}. Then, again by Proposition 2, {\mathcal{P}}_{E} is minimal, as required. □
Example 2 Let p=3 and \mathcal{M}=\left[\begin{array}{cc}1& 0\\ 0& 2\end{array}\right] with determinant 2. Thus {\mathcal{M}}^{7}\equiv \mathcal{M}mod3 holds, and the relators of the semidirect product presentation, as in (4), will be {y}_{1}{y}_{2}={y}_{2}{y}_{1}, {x}^{7}=x, {y}_{1}x=x{y}_{1} and {y}_{2}x=x{y}_{2}^{2}. By Theorem 3, the simplest (unique) characteristic polynomial {\nu}^{2}3\nu +2 represents this inefficient but minimal presentation. A similar example can also be obtained by considering the matrix \left[\begin{array}{cc}2& 0\\ 0& 1\end{array}\right].
Conjecture 1 Let us consider a 2\times 2matrix ℳ with eigenvalues {\lambda}_{1} and {\lambda}_{2}, respectively. Then, by considering tr(\mathcal{M})={\lambda}_{1}+{\lambda}_{2} and det(\mathcal{M})={\lambda}_{1}{\lambda}_{2}, one can investigate whether the minimality conditions of Theorem 3 can be expressed as a unique statement
gcd(tr(\mathcal{M}),det(\mathcal{M}))=1.
2.3 Part III: Array polynomials over {\mathcal{P}}_{E}
In this section, we are mainly interested in the generating functions (in terms of array polynomials) related to the presentations defined in (2) and (4). In [13, 19], by considering the generating pictures in two different group and monoid extensions, the authors have investigated the related generating functions over the presentations of them. However, in a different manner, here we will investigate the array polynomials (as generating functions) in the meaning of characteristic polynomials obtained in the previous section. In other words, by taking into account Proposition 3 and Theorems 1, 2 and 3, we will reach our aim using semidirect products of monoids studied in this paper.
As noted in [[13], Remark 1.1], if a monoid presentation satisfies efficiency or inefficiency (while it is minimal), then it always contains a minimal number of generators. Working with a 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 (2) with array polynomials. In fact, array polynomials {S}_{a}^{n}(x) are defined by means of the generating function
\frac{{({e}^{t}1)}^{a}{e}^{tx}}{x!}=\sum _{n=0}^{\mathrm{\infty}}{S}_{a}^{n}(x)\frac{{t}^{n}}{n!}
(cf. [20–22]). According to the same references, array polynomials can also be defined as follows:
{S}_{a}^{n}(x)=\frac{1}{a!}\sum _{j=0}^{a}{(1)}^{aj}\left(\begin{array}{c}a\\ j\end{array}\right){(x+j)}^{n}.
(10)
Since the coefficients of array polynomials are integers, they find a very large application area, especially in system control (cf. [23]). In fact, these integer coefficients give us an opportunity to use the polynomials in our case. We should note that there also exist some other polynomials, namely Dickson, Bell, Abel, MittagLeffler etc. having integer coefficients which will not be handled in this paper. Now, by using (10) and keeping in our mind that the coefficients of array polynomials are integers, we clearly have
{S}_{a}^{n}(b)=\{\begin{array}{ll}{b}^{n};& a=0,\\ b;& a=0\text{and}n=1,\\ 1;& a=n\text{or}n=a=0.\end{array}
(11)
Therefore we have the following theorem.
Theorem 4 The presentation {\mathcal{P}}_{E} defined in (2) has a set of generating functions {p}_{1}(x)=[{S}_{0}^{k}(x)][{S}_{0}^{l}(x)] and
{p}_{2}({y}_{i})={S}_{0}^{1}(x)+[{S}_{0}^{{\alpha}_{im}}({y}_{1})]+[{S}_{0}^{{\alpha}_{im}}({y}_{2})]+\frac{1}{{S}_{0}^{1}(x)},
where {S}_{a}^{n}(x) and {S}_{a}^{n}({y}_{i}) are defined as in (11), {\alpha}_{im}’s are the entries of ℳ and 1\le i,m\le 2.
Proof Let us consider the presentation {\mathcal{P}}_{E} in (2). We recall that for {\mathcal{P}}_{E} to be a semidirect product presentation of the free abelian monoid K rank two by cyclic monoid A of order k, by (1), the power matrices {\mathcal{M}}^{k} and {\mathcal{M}}^{l} (1\le l<k and l,k\in {\mathbb{Z}}^{+}) must be congruent to each other by modulo d, where d\mid kl. By the meaning of endomorphism, this congruence actually comes from the existence of the relator
where x is the unique generator of A. In this relator, by replacing x by an array polynomial {S}_{0}^{1}(x) and also considering the status modulo d, we can get {p}_{1}(x) as a generating function. Basically, the function {p}_{1}(x) must exist in all such sets that contain generating functions of the presentation {\mathcal{P}}_{E} in (2).
Now, let us give our attention to investigating the existence of generating functions of the form {p}_{2}({y}_{i}). Actually, the existence of these functions is again based on the existence of the equality in (1). Because the relators
{y}_{1}x=x{y}_{1}^{{\alpha}_{11}}{y}_{2}^{{\alpha}_{12}}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{y}_{2}x=x{y}_{1}^{{\alpha}_{21}}{y}_{2}^{{\alpha}_{22}}
have been obtained by the meaning of endomorphism on the base matrix ℳ, it is clearly seen that while the first row of ℳ gives the above first composite relator, the second row gives the second one. Thus, by replacing {y}_{1} by the array polynomial {S}_{0}^{1}({y}_{1}) and {y}_{2} by the array polynomial {S}_{0}^{1}({y}_{2}) into the above relators, we get the required functions. Notice that although in monoids we are not allowed to apply the inverse element {x}^{1}, by the meaning of functions, we can use this inverse element as a fractal of the form \frac{1}{{S}_{0}^{1}(x)}.
Notice also that in the light of the above material, for the relator {y}_{1}{y}_{2}={y}_{2}{y}_{1}, the generating function will only be the zero function. □
Nevertheless, by considering Theorem 1 and Equation (8), we can also get the following result.
Theorem 5 The presentation {\mathcal{P}}_{E} defined in (2) has a single generating function
p(\nu )={S}_{0}^{2}(\nu )({\lambda}_{1}+{\lambda}_{2}){S}_{0}^{1}(\nu )+({\lambda}_{1}{\lambda}_{2}){S}_{n}^{n}(\nu ),
(12)
where ν represents threeordered variables (x,{y}_{1},{y}_{2}) and {\lambda}_{1}, {\lambda}_{2} are the eigenvalues of ℳ.
Proof Since, by Theorem 1, the presentation {\mathcal{P}}_{E} can be represented by a unique characteristic polynomial as in (8), the function obtained from this polynomial should be the unique generating function. However, since this unique function must represent the whole presentation, it should be constructed by all generators of {\mathcal{P}}_{E}. In other words, the required function should be defined as multivariable (x,{y}_{1},{y}_{2}). Hence, by replacing variables by array polynomials as previously, we can get the generating function as given in (12). □
Conjecture 2 By taking into account Theorems 1, 4 and 5, one can investigate whether there exists an equivalence between the generating functions that represent the presentation {\mathcal{P}}_{E} in (2) as indicated in Figure 2.
By considering Proposition 1, Theorem 2 and Theorem 4, respectively, we further obtain the following corollary.
Theorem 6 Suppose {\mathcal{P}}_{E} in (2) is an efficient presentation. Then it accepts
{p}_{3}(x)=[{\lambda}_{1}{\lambda}_{2}1]{S}_{0}^{1}(x)
as a generating function including the functions defined in Theorem 4, where {S}_{a}^{n}(x) is defined as in (11) and {\lambda}_{1}, {\lambda}_{2} are the eigenvalues of ℳ.
Proof By Theorem 4, we have the generating functions {p}_{1}(x) and {p}_{2}({y}_{i}) (1\le i\le 2). However, since {\mathcal{P}}_{E} is given as an efficient presentation, we also need to add a function related to its efficiency. At this stage, we can think about Theorem 2 which basically says that {\mathcal{P}}_{E} is efficient if {\lambda}_{1}{\lambda}_{2}1\equiv 0modp. Moreover, this condition is directly related to the generator x in {\mathcal{P}}_{E} (cf. [[9], Theorem 2.4]). Therefore this last generating function should be of the form [{\lambda}_{1}{\lambda}_{2}1]{S}_{0}^{1}(x), as required. □
The following corollary is a consequence of Theorem 5 which states that the expression of Theorem 6 can also be given as a single condition.
Corollary 1 Suppose {\mathcal{P}}_{E} in (2) is an efficient presentation. Then it accepts p(\nu )={S}_{0}^{2}(\nu )({\lambda}_{1}+{\lambda}_{2}){S}_{0}^{1}(\nu )+1 as a generating function, where ν represents threeordered variables (x,{y}_{1},{y}_{2}) and {\lambda}_{1}, {\lambda}_{2} are the eigenvalues of ℳ.
The sketch of proof Since the efficiency of a presentation implies {\lambda}_{1}{\lambda}_{2}\equiv 1modp for any prime p or 0, we can take {\lambda}_{1}{\lambda}_{2}=1 in (12) because, by NFT, the simplest (reduced) form represents all other generating functions p(\nu ). Thus, depending on the constants, the function p(\nu ) in the result can be thought to be in the congruence class of all such functions. □
Remark 1 From Corollary 1 and by considering Theorem 1, we can easily deduce that the set of generating functions for an efficient presentation can be presented as a single element. This actually shows the importance of efficiency during the study of generating functions.
In Proposition 2 and also in Theorem 3, we expressed the minimality (while satisfying inefficiency) of the presentation {\mathcal{P}}_{E} in (4) in two different versions, respectively. As a next step of Theorem 6 and as a consequence of Theorem 4, we will deal with the array polynomials which are obtained from a minimal but inefficient presentation. The following lemma will be needed in the proof of our next result.
Lemma 1 There always exists {(p1)}^{2p}\equiv 1modp for any prime p.
Proof We first note that the lemma is clear for p=2. So, let us assume that p is an odd prime.
Suppose that {(p1)}^{2p}\equiv 0modp. So, by the meaning of congruence, we must have {(p1)}^{2p}=pt for a t\in {\mathbb{Z}}^{+}. In fact, the lefthand side of this equality can be written as a binomial sum
{(p1)}^{2p}={p}^{2p}2{p}^{2p}+(2{p}^{2}p)\cdot {p}^{2(p1)}\cdots +1.
In this sum, each term except the last one, which is 1, is congruent to 0 by modulo p since each of them contains p as a factor. Thus, {(p1)}^{2p} cannot be congruent to 0 by modulo p unless we add −1 to both sides of this congruence. Therefore {(p1)}^{2p}\equiv 1modp, as required. □
Theorem 7 For an odd prime p, the minimal but inefficient presentation
{\mathcal{P}}_{E}=[{y}_{1},{y}_{2},x;{y}_{1}{y}_{2}={y}_{2}{y}_{1},{x}^{2p+1}=x,{y}_{1}x=x{y}_{1}^{p1},{y}_{2}x=x{y}_{2}]
has a set of generating functions
\begin{array}{c}{p}_{1}(x)=[{S}_{0}^{2p+1}(x)][{S}_{0}^{1}(x)],\phantom{\rule{2em}{0ex}}{p}_{2}({y}_{1})={S}_{0}^{1}(x)+[{S}_{0}^{p1}({y}_{1})]+\frac{1}{{S}_{0}^{1}(x)},\hfill \\ {p}_{3}(x)=[{\lambda}_{1}{\lambda}_{2}p+1]{S}_{0}^{1}(x),\phantom{\rule{2em}{0ex}}{p}_{2}({y}_{2})={S}_{0}^{1}(x)+[{S}_{0}^{1}({y}_{2})]+\frac{1}{{S}_{0}^{1}(x)},\hfill \end{array}
where {S}_{a}^{n}(x) and {S}_{a}^{n}({y}_{i}) are defined as in (11).
Proof Let p be an odd prime. We first need to show that {\mathcal{P}}_{E} in the theorem actually presents a semidirect product of a free abelian monoid rank two by a finite cyclic monoid as defined in (2). To do that, we just have to ensure that Equation (1) holds.
From {\mathcal{P}}_{E}, one can easily obtain a diagonal matrix \mathcal{M}=\left[\begin{array}{cc}p1& 0\\ 0& 1\end{array}\right] which det\mathcal{M}={\lambda}_{1}{\lambda}_{2}=p1 for eigenvalues {\lambda}_{1} and {\lambda}_{2}. Moreover, by Lemma 1, we obviously get {(p1)}^{2p+1}\equiv p1modp similarly as the other 2p+1th powers of entries. Thus we obtain
{\mathcal{M}}^{2p+1}\equiv \mathcal{M}modp,
which implies that {\mathcal{P}}_{E} actually presents the required semidirect product. Hence, as indicated in the proof of Theorem 6, we have the generating functions {p}_{1}(x) and {p}_{2}({y}_{i}) for 1\le i\le 2 (by Theorem 4).
However, again as in the proof of Theorem 6, we need to find a new function (array polynomial) which is related to the minimality (having inefficiency) status of {\mathcal{P}}_{E}. To do that, we will take into account the same way which was obtaining the function {p}_{3}(x)=[{\lambda}_{1}{\lambda}_{2}1]{S}_{0}^{1}(x) in Theorem 6. From Proposition 1, we know that any semidirect product presentation is not efficient if the determinant of the base matrix ℳ is equivalent to 0 or p1 by modulo p, and also by Proposition 2, this presentation is minimal (whilst it is efficient) if the prime p is odd and the entries of the main diagonal are p1 and 1 (or vice versa) in the diagonal matrix. It is clear that all these situations are suitable for our case in this proof. Therefore, since we have {\lambda}_{1}{\lambda}_{2}=p1, there must exist [{\lambda}_{1}{\lambda}_{2}(p1)]{S}_{0}^{1}(x) as a generating function.
Hence the result. □
By Theorem 5, one can express Theorem 7 as in the following corollary. In fact, this result is the minimality version of Corollary 1. We will again omit the proof since it is quite clear by taking {\lambda}_{1}{\lambda}_{2}=p1 and {\lambda}_{1}+{\lambda}_{2}=tr(\mathcal{M})=p in (12).
Corollary 2 Let us consider the minimal but inefficient presentation {\mathcal{P}}_{E} in Theorem 7. Then it accepts p(\nu )={S}_{0}^{2}(\nu )p{S}_{0}^{1}(\nu )+p1 as a generating function, where ν represents threeordered variables (x,{y}_{1},{y}_{2}).
Similarly to Corollary 1, the function p(\nu ) in the above corollary can be taken as the simplest (reduced) form being in the same congruence class as all other generating functions p(\nu ) that represent this minimal presentation.
Remark 2 From Corollary 2 and by considering Theorem 1, we can easily deduce that the set of generating functions for a minimal but inefficient presentation can be presented as a single element. Therefore, as pointed out in Remark 1, this shows the importance of minimality (even holding inefficiency) in the meaning of the study of generating functions.
As a result of Theorem 5 and Corollaries 1, 2, we can state that array polynomials obtained in this paper are congruent to each other and so they construct congruence classes. Since these results are based on the characteristic polynomials, the proof of the following main theorem of this paper can be seen quite similar to the proofs of Proposition 3 and Theorem 1.
Theorem 8 Each array polynomial obtained from the presentation {\mathcal{P}}_{E} in (2) appears to be a congruence class. Moreover, this presentation is represented by a single type of array polynomials depending on this congruence. This single type may contain a unique congruence class of array polynomials in the case of efficiency or minimality status of {\mathcal{P}}_{E}.