Semilattice of topological groups

Abstract

In this article, we establish necessary and sufficient condition on a topological Clifford semigroup to be a semilattice of topological groups. As a consequence, we show that a topological Clifford semigroup satisfies the property that for each and every there exists an element such that if and only if it is a strong semilattice of topological groups if and only if it is a semilattice of topological groups. We prove that some topological properties like regularity and completely regularity are equivalent in a semilattice of topological groups. We also prove that the quotient space of a semilattice of topological groups by a full normal Clifford subsemigroup is again a semilattice of topological groups. Finally, we establish that if is a family of semilattices of topological groups and Ni is a full normal Clifford subsemigroup of Si for all then is topologically isomorphic to