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 $(S,\tau )$ satisfies the property that for each $G\in \tau$ and every $x\in G,$ there exists an element $U\in \tau$ such that $x\in U\subseteq G\cap J_x$ 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 $T_0,T_1,T_2,$ 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 $\{S_i:i=1,2,\dots ,n\}$ is a family of semilattices of topological groups and N_i is a full normal Clifford subsemigroup of S_i for all $i=1,2,\dots ,n,$ then ${\otimes }_{i=1}^n(S_i/N_i)$ is topologically isomorphic to ${\otimes }_{i=1}^n{S}_i/{\otimes }_{i=1}^n{N}_i.$

KEYWORD:

• Semilattice
• semilattice of topological groups
• topological Clifford semigroup
• topological group
• topological semigroup

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 22A15
• 54E35
• 54E18
• 54D30

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.

Additional information

Funding

The research of the second author was supported by Council for Scientific and Industrial Research, India. CSIR Award no. : 09/028(1001)/2017-EMR-1.