On the endomorphism monoids of Clifford semigroups

Abstract

In this paper, we study properties of the endomorphism monoids of strong semilattices of groups. In Sec. 2, several properties for endomorphism monoids of finite semilattices are investigated. In Sec. 3, we collect some results on endomorphism monoids of strong semilattices of groups, i.e. Clifford semigroups.

Endomorphisms of Clifford semigroups with injective structure homomorphisms

“…On the other hand, in [11], the author constructs an endomorphism on S based on an endomorphism on G ν . This construction does not require that the structure homomorphisms are injective.…”

“…Proposition 2. [11] Let Y be a semilattice which has a least element ν = Y and let S = ξ∈Y G ξ be a Clifford semigroup. For g ∈ End(G ν ), we define f : S → S by (x ξ )f = (x ξ )f ξ , where…”

“…Let note that the monoid End(S) is related to the monoid End(Y ) of all endomorphisms on the semilattice Y . In fact, if End(S) is regular (idempotent and completely regular, respectively) then End(Y ) is also regular (idempotent and completely regular, respectively) by Theorem 3.10 in [11]. We will use this fact subsequently without referring it.…”

“…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].…”

“…We will use this fact subsequently without referring it. In [11], the author studied the endomorphisms on a semilattice Y . In particular, the author has characterized the semilattices Y with idempotent and completely regular, respectively, endomorphism monoid End(Y ).…”