Complete Invariant ⋆-Metrics on Semigroups and Groups

Abstract

In this paper, we study the complete ⋆-metric semigroups and groups and the Raǐkov completion of invariant ⋆-metric groups. We obtain the following. (1) Let (X,d⋆) be a complete ⋆-metric space containing a semigroup (group) G that is a dense subset of X. If the restriction of d⋆ on G is invariant, then X can become a semigroup (group) containing G as a subgroup, and d⋆ is invariant on X. (2) Let (G,d⋆) be a ⋆-metric group such that d⋆ is invariant on G. Then, (G,d⋆) is complete if and only if (G,τd⋆) is Raǐkov complete.

Keywords

• ⋆-metric
• topological group
• topological semigroup
• Raǐkov completion