Non-commutative finite monoids of a given order n ≥ 4

Abstract

Abstract For a given integer n = p 1 α 1 p 2 α 2 ⋯ p k α k $n = p_1^{\alpha _1 } p_2^{\alpha _2 } \cdots p_k^{\alpha _k }$ (k ≥ 2), we give here a class of finitely presented finite monoids of order n. Indeed the monoids Mon(π), where π = 〈 a 1 , a 2 , … , a k | a i p i α i = a i ,   ( i = 1 , 2 , … , k ) , a i a i + 1 = a i ,   ( i = 1 , 2 , … , k − 1 ) 〉 . $$\pi = {\langle {a_1 ,a_2 , \ldots ,a_k |a_i^{p_i^{\alpha _i } } = {a_i}, {\left({i = 1,2, \ldots ,k} \right)}, a_i a_{i + 1} = {a_i}, \left({i = 1,2, \ldots ,k - 1} \right)} \rangle} .$$ As a result of this study we are able to classify a wide family of the k-generated p-monoids (finite monoids of order a power of a prime p). An interesting di erence between the center of finite p-groups and the center of finite p-monoids has been achieved as well. All of these monoids are regular and non-commutative.