On the identification of finite non-group semigroups of a given order

Identifying finite non-group semigroups for every positive integer is significant because of many applications of such semigroups are functional in various branches of sciences such as computer science, mathematics and finite machines. The finite non-commutative monoids as a type of such semigroups were identified in 2014, for every positive integer. We here attempt to identify the finite commutative monoids and finite commutative non-monoids of a given integer n=pαqβ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=p^\alpha q^\beta$$\end{document}, for every integers α,β≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha , \beta \ge 2$$\end{document} and different primes p and q. In order to recognize the commutative monoids, we present a class of 2-generated monoids of a given order, and for the commutative non-monoids of order n=pαqβ,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=p^\alpha q^\beta,$$\end{document} we give the minimal generating set. Moreover, we prove that there are exactly (pα-2)(qβ-2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p^{\alpha }-2)(q^{\beta }-2)$$\end{document} non-isomorphic commutative non-monoids of order pαqβ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p^\alpha q^\beta$$\end{document}. The identification of non-group semigroups for the integers p2α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p^{2\alpha }$$\end{document} and 2pα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2p^\alpha$$\end{document} is achieved. The automorphism groups of these groups are specified as well. As a result of this study, an interesting difference between the abelian groups and the commutative semigroups of order p2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p^2$$\end{document} is presented.


Introduction
Identifying a finite non-group semigroup (commutative or non-commutative) of order of a given positive integer n could be significant because of the interesting applications of finite semigroups in computer science, mathematics and finite machines. As a subclass of non-group semigroups, the non-commutative monoids of a given order were identified in 2014 by Ahmadi et al. [2] to show that for every positive integer n ≥ 4, there exists a non-commutative monoid of order n. In this paper, we intend to identify two types of finite semigroups, commutative non-monoids of order p q and commutative monoids of order of a given positive integer n ≥ 4 . By giving the unique minimal generating set for the commutative non-monoids, we specify the number of all non-isomorphic semigroups of this type. Subsequently, we present a class of finitely presented non-isomorphic commutative monoids of a given positive integer n.
Our notation is merely standard, and we follow [3][4][5][6][7]. The detailed information on semigroup (or monoid) presentations may be found in [3,4,7]. These articles investigate the efficiency of finitely presented semigroups and the related monoids. We prefer to give a brief history on the finitely presented semigroups and monoids. A semigroup (or monoid) S is said to be presented by a semigroup (or monoid) pres- . As usual, we will use the notation Sg( ) for the semigroup presented by the presentation = ⟨A | R⟩ . Also, the deficiency of a finite semigroup presentation = ⟨A | R⟩ is defined to be | R | − | A | and is denoted by def( ) . The semigroup deficiency def s (S) of a finitely presented semigroup S is given by as well.
For every positive integers m, r and s, we define the presentations: and let S m,r = Sg( m,r ) . Obviously, the semigroup S m,1 is the cyclic group of order m − 1 ; however, S m,r is a non-monoid monogenic semigroup of order m + r − 1 , for every positive integer r ≥ 2 . These last monogenic semigroups will be used in construction of the commutative non-monoid semigroups. By using the well-known definition of direct product of semigroups [8], we may easily check that the direct product of two non-monoid monogenic semigroups is also a non-monoid semigroup. Note that, the direct product of semigroups S m,1 and S m ′ ,1 , as studied in [1], is a 2-generated semigroup. However, the situation is completely different in the direct product of non-monoid monogenic semigroups as we will see in the following sections.

Main results
As a preliminary result, we have the following lemma on the monogenic semigroups. Proof Let T 1 = S m,r and T 2 = S m � ,r � be two non-monoids of orders p and q , respectively. So, m + r = p + 1 and m � + r � = q + 1 . Let S = T 1 × T 2 be the direct product of T 1 and T 2 . Every element of S is an ordered pair (a i , b j ), where 1 ≤ i ≤ p and 1 ≤ j ≤ q . Since T 1 and T 2 are commutative non-monoid semigroups, it may be easily checked that S is also a commutative non-monoid semigroup. Then, by the m,r = ⟨a | a m+r = a r ⟩, results of Lemma 2.1 and the definition of S as above, we conclude that there are exactly (p − 2)(q − 2) number of commutative non-monoid semigroups of order p q . To complete the proof, we have to give the minimal generating set for S. Without lose of generality, suppose that p < q . First of all by considering the elements of S, we may easily check that every element c ij = (a i , b j ) of S may be presented as: Remind the relators a p +1 = a r and b q +1 = b s where r, s ≥ 2, then, any product yz of the elements of X belongs to S∖X . Also, any product yz of the elements of S∖X belongs to S∖X . Consequently, S�S 2 = X . ◻ To identify the commutative monoids of a given integer n ≥ 4, we show that the disjoint union of two finite cyclic groups H = ⟨a⟩ and K = ⟨b⟩ becomes a commutative non-group monoid if the multiplication is defined as ab = a = ba.  = (a, b),

3
for every i = 1, 2, … , r . So, Sg( � r,s ) is a commutative monoid. Although Sg( � r,s ) is the disjoint union of two cyclic groups generated by a and b, it is not a group, because the element a has not the group inverse in Sg( � r,s ). The number of different pairs (r, s) of the integers r, s ≥ 2 satisfying the equation n = r + s , is equal to n − 3 . Also, it is not very hard to see that the semigroups Sg( � r,s ) and Sg( � r � ,s � ) are non-isomorphic for any two different pairs (r, s) and (r � , s � ) . Hence there are exactly n − 3 non-isomorphic commutative monoids of order n. ◻

Non-isomorphic semigroups
Two special cases of non-group commutative semigroups of orders 2p and p 2 are of interest to consider, for every odd prime p and every integer ≥ 2 . Our results on the identification of semigroups are collected in the following theorems.
commutative non-monoid semigroups of order p 2 , for every odd prime p. This is a substantial difference between the groups and semigroups of order p 2 .

Conclusion
As an example we will give an efficient presentation for the commutative non-monoid semigroup of order 3 2 . Before presenting this example, we have a look at the automorphism groups of the commutative non-monoid semigroups in the following theorem. Proof To prove (i), observe that any automorphism ∈ Aut(S m,r ) , where r ≥ 2 , has to map a generator to a generator. Since S m,r possesses a unique generating element, is the identity map.
For (ii), let S = S m,r × S m,r where m + r − 1 = p . We may define the involution automorphism ∈ Aut(S) by (a i , b j ) = (a j , b i ) , where 1 ≤ i, j ≤ p + 1 . By considering the generating set we get (x) = x , (A i ) = B i and (B i ) = A i , for every i = 2, 3, … , p . Hence, we have to verify the relators: The proofs are straightforward because of the commutativity of S and the key relators A i A j = xA i+j−1 , B i B j = xB i+j−1 and Finally, consider the subgroups H = {a, a 2 , … , a r } and K = {b, b 2 , … , b s } of the semigroup Sg( � {r,s} ) . For any ∈ Aut(Sg( � r,s )) , the restrictions | H and | K are automorphisms of H and K, respectively. As a well-known result of the cyclic groups, there are exactly (r) number of generating elements for H, and then, | Aut(H) |= (r) . Similarly, | Aut(K) |= (s) . Since the cross-product of two automorphisms is an automorphism, (iii) follows at once. ◻ Obtaining a finite presentation for the non-group commutative semigroups of order n = p q requires too long and tedious hand calculation. As an example, we give here a finite presentation for the non-group commutative non-monoid semigroup of order p 2 where p = 3 . Let T = Sg(S 2,2 × S 2,2 ) . Then, T is of order 9 and possesses an efficient presentation as in the following example.

Example 4.2
The semigroup T may be presented as ⟨X | R⟩ where X = {x, 1 , 2 , 3 , 4 } and R is the set of 29 relators: To verify these relators, one may use the original definition of the minimal generating set X. Indeed, by letting 3 ) , all of the products of two elements of X could be calculated by using Lemma 2.1. Then, the redundant relators have to be deleted. By using Gap code [9], we now believe that this presentation is an efficient semigroup presentation for T.
This example along with our computations results the following conjecture on the non-group commutative nonmonoid semigroups of order p 2 .
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