On Endomorphisms of Semilattices of Groups

Abstract

The set of all endomorphisms of an algebraic structure with composition of functions as operation is a rich source of semigroups, which has only rarely been dipped into (see [2]). Here we make a start by considering endomorphisms of Clifford semigroups, relating them to the homomorphisms and endomorphisms of the underlying groups.

Endomorphisms of Clifford semigroups with injective structure homomorphisms

“…We start with three propositions which include some already known results. In [10], it is shown that any f ∈ End(S) is determined by f ν . Proposition 1.…”

“…Proposition 1. [10] Let Y be a semilattice which has a least element ν = Y and let S = ξ∈Y G ξ be a Clifford semigroup with injective structure homomorphisms…”

“…The image of a homomorphism f ∈ Hom(S, T ) is denoted by Im(f ) = {(x)f : x ∈ S}. Endomorphisms on Clifford semigroups were studied for examples in [10,11].…”

“…Let us consider a Clifford semigroup S = ξ∈Y G ξ and let f ∈ End(S). In [10], the authors show that the restriction of an endomorphism f on S to the set E S = {e ξ : ξ ∈ Y } of the idempotent elements in S induces an endomorphism on Y , which we denote by f . In fact, (α)f = β, whenever (e α )f = e β .…”

“…We suppose that the structure homomorphisms are injective and the semilattice Y has a least element ν. Then an endomorphism f on S is already determined by the endomorphism f ν ∈ Hom(G ν , G (ν)f ) [10]. We will study the endomorphism monoid of such a Clifford semigroup S. In particular, we consider the case that End(S) is regular and completely regular, respectively.…”