Non-commutative finite monoids of a given order n >= 4

For a given integer n = p(1)(alpha 1) p(2)(alpha 2) ... p(k)(alpha k) (k >= 2), we give here a class of finitely presented finite monoids of order n. Indeed the monoids Mon(pi), where pi = < a(1), a(2), ... , a(k)vertical bar a(i)(pi alpha i) = a(i), (i = 1, 2, ... , k), a(i)a(i+1) (i = 1, 2, ... , k - 1)>. 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 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.