COMPLETE INVARIANT FUZZY METRICS ON SEMIGROUPS AND GROUPS

In this paper, we study the Raǐkov completion of invariant fuzzy metric groups and complete fuzzy metric semigroups (in the sense of Kramosil and Michael). We establish that: (1) if $ (G, M,\ast) $ is a fuzzy metric group such that $ (M,\ast) $ is invariant, then the Raǐkov completion $ \varrho G $ of $ (G,\tau_{M}) $ is a fuzzy metric group $ (\varrho G, \widetilde{M},\ast) $ such that $ (\widetilde{M},\ast) $ is invariant on $ \varrho G $ and $ \widetilde{M}_{|G\times G\times [0,\infty)} = M $; (2) if $ (G, M, \ast) $ is a fuzzy metric semigroup such that $ (M, \ast) $ is invariant, then a fuzzy metric completion $ (\widetilde{G}, \widetilde{M}, \ast) $ of $ (G, M, \ast) $ is a fuzzy metric semigroup and $ (\widetilde{M}, \ast) $ is invariant.