Presentation of semidirect product of monogenic semigroups

Abstract

By studying the semigroup presented by π = A, B | A n+1 = A, B m+1 = B, BA = A n−1 B, B m = A n , for every positive integers m, n ≥ 3 we show that, for even values of m this is an appropriate presentation for the semidirect product of monogenic semigroup S = a | a n+1 = a by the monogenic semigroup T = b | b m+1 = b. Moreover, for odd values of m it is a commutative semigroup of order (3+(−1) n)m 2. The interests of presentability of products of semigroups are quite useful in the study of more effective and intrinsic properties of the products (like as the computation of their characters and studying their Green J-classes). Our method of proof based on using the theoretic definitions and hereby we will deduced the presentation of the direct product of monogenic semigroups for every integers n, m ≥ 3.