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

For a given integer n = p1 1 p α2 2 . . . p αk k , (k ≥ 2), we give here a class of finitely presented finite monoids of order n. Indeed the monoids Mon(π), where π = 〈a1, a2, . . . , ak|a p αi i i = ai, (i = 1, 2, . . . , k), aiai+1 = ai, (i = 1, 2, . . . , k − 1)〉. As a result of this study we are able to classify a wide family of the kgenerated p-monoids (finite monoids of order a power of a prime p). An interesting difference 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.