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

Abstract

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=p^alpha q^beta, for every integers alpha , beta ge 2 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=p^alpha q^beta, we give the minimal generating set. Moreover, we prove that there are exactly (p^{alpha }-2)(q^{beta }-2) non-isomorphic commutative non-monoids of order p^alpha q^beta. The identification of non-group semigroups for the integers p^{2alpha } and 2p^alpha 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 p^2 is presented.