On continuity of homomorphisms between topological Clifford semigroups

Abstract

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.