Endomorphisms of Clifford semigroups with injective structure homomorphisms

In the present paper, we study semigroups of endomorphisms on Clifford semigroups with injective structure homomorphisms, where the semilattice has a least element. We describe such Clifford semigroups having a regular endomorphism monoid. If the endomorphism monoid on the Clifford semigroup is completely regular then the corresponding semilattice has at most two elements. We characterize all Clifford semigroups Gα∪Gβ (α>β) with an injective structure homomorphism, where Gα has no proper subgroup, such that the endomorphism monoid is completely regular. In particular, we consider the case that the structure homomorphism is bijective.

On the endomorphism monoids of Clifford semigroups

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.

The Strong Semilattice of Pi-groups

A semigroup is called a GV-inverse semigroup if and only if it is isomorphic to a semilattice of $\pi$-groups. In this paper, we give the sufficient and necessary conditions for a GV-inverse semigroup to be a strong semilattice of $\pi$-groups. Some conclusions about Clifford semigroups are generalized.

Takahasi semigroups

Takahasi’s Theorem on chains of subgroups of bounded rank in a free group is generalized to several classes of semigroups. As an application, it is proved that the subsemigroups of periodic points are finitely generated and periodic orbits are bounded for arbitrary endomorphisms for various semigroups. Some of these results feature classes such as completely simple semigroups, Clifford semigroups or monoids defined by balanced one-relator presentations.

On continuity of homomorphisms between topological Clifford semigroups

Generalizing an old result of Bowman we prove that a homomorphism $f:X\to Y$ between topological Clifford semigroups is continuous if the idempotent band $E_X=\{x\in X:xx=x\}$ of $X$ is a $V$-semilattice;the topological Clifford semigroup $Y$ is ditopological;the restriction $f|E_X$ is continuous;for each subgroup $H\subset X$ the restriction $f|H$ is continuous.