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
Acknowledgement
The authors are grateful to the Learned Referee for several useful comments, suggestions and providing this proof of Theorem 3.1 which have definitely enriched the article.