For a monoid M and a subsemigroup S of the full transformation semigroup T n , the wreath product M ≀ S is defined to be the semidirect product M n ⋊ S, with the coordinatewise action of S on M n . The full wreath product M ≀ T n is isomorphic to the endomorphism monoid of the free M -act on n generators. Here, we are particularly interested in the case that S = Sing n is the singular part of T n , consisting of all non-invertible transformations. Our main results are presentations for M ≀Sing n in terms of certain natural generating sets, and we prove these via general results on semidirect products and wreath products. We re-prove a classical result of Bulman-Fleming that M ≀ Sing n is idempotent-generated if and only if the set M/L of L -classes of M forms a chain under the usual ordering of L -classes, and we give a presentation for M ≀ Sing n in terms of idempotent generators for such a monoid M . Among other results, we also give estimates for the minimal size of a generating set for M ≀ Sing n , as well as exact values in some cases (including the case that M is finite and M/L is a chain, in which case we also calculate the minimal size of an idempotent generating set). As an application of our results, we obtain a presentation (with idempotent generators) for the idempotent-generated subsemigroup of the endomorphism monoid of a uniform partition of a finite set.

Let G be a finite group and X be a G-conjugacy of elements of order 3. The A4-graph of G is a simple graph with vertex set X and two vertices x,yÎX are linked if x≠ y and xy-1 is an involution element. This paper aims to investigate  the A4-graph properties for the monster  Held group He.

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It is well known that the wreath product  is the endmorphism monoid of a free S-act with n-generators. If S is a trivial semigroup then  is isomorphic to . The extension  for  to  where  is an independent algebra has been investigated. In particular, we consider  is to be , where  is a free left S-act of n-generators. The eventual goal of this paper is to show that  is an endomorphism monoid of a free left S-act of n-generators and to prove that  is embedded in the wreath product .