On Presentation of Total Wreath Product

Abstract

     When M is any monoid and  T_n  is a total transformation semigroup that consists of all maps from a set of cardinalities n to itself such that n∈N, the total wreath product M ≀_n T_n  is investigated to be  〖M 〗^n ⋊_n T_n, a semidirect product of 〖M 〗^n  by T_n. This paper is particularly interested in the case that M is a finite group G. Clearly, the total wreath product G ≀_n T_n  is a monoid that is isomorphic to the endomorphism monoid ENDF_n (G ), where F_n (G ) is an independent algebra and F_n (G )=∪ ̇_(j=1)^n Gy_j   is a free left G-acts on  n generators. During this article, it was shown and investigated that the total wreath product  G ≀_n T_n  is equal to a semidirect product of 〖G 〗^n  by T_n   〖( 〖G 〗^n ⋊_n T〗_n), and expressed a presentation for G ≀_n T_n   with respect to a minimal generating set of 〖〖G 〗^n ⋊_n T〗_n  for an application of our results.