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complete invariant fuzzy metrics on semigroups and groups
Li-Hong Xie
Keywords:Fuzzy metric; topological group; topological semigroup; Ra\v{\i}kov completion
Abstract:
      In this paper, we study the Ra\v{\i}kov 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 Ra\v{\i}kov 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.