On induced $L$−fuzzy uniformities
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The extension of the uniformity on the family of fuzzy subsets $L^Y$ to a uniformity on the family of fuzzy subsets $L^X : Y \subseteq X$ and the restriction of the uniformity on $L^Y$ to a uniformity on $L^Y$ are defined and studied. The induced (fuzzy) quasi-uniformity on $P^∗(L)^X$ for each given (fuzzy) quasi-uniformity on $L^X$ is defined. Moreover, the induced $(P^∗(L), M)$−fuzzy quasi-uniformity on $P^∗(L)^X$, for each $(L, M)$−fuzzy quasi-uniformity on $L^X$ is studied. In each case, the relation between their interior operators is obtained. Finally, the relation between the category $\bf Qunif(L,M)$ of all $(L, M)$−fuzzy quasi uniformity spaces and all quasiuniformly continuous functions, and the category $\bf FQunif(P^∗(L),M)$ of all $(P^∗ (L), M)$−fuzzy quasi uniformity spaces and quasi-uniformly continuous fuzzy functions is outlined. It is remarked that all kinds of categories of quasi-uniform spaces and quasi-uniformly continuous functions can be derived from the category $\bf FQunif(P^∗(L),M)$.
Abstract
The extension of the uniformity on the family of fuzzy subsets $L^Y$ to a uniformity on the family of fuzzy subsets $L^X : Y \subseteq X$ and the restriction of the uniformity on $L^Y$ to a uniformity on $L^Y$ are defined and studied. The induced (fuzzy) quasi-uniformity on $P^∗(L)^X$ for each given (fuzzy) quasi-uniformity on $L^X$ is defined. Moreover, the induced $(P^∗(L), M)$−fuzzy quasi-uniformity on $P^∗(L)^X$, for each $(L, M)$−fuzzy quasi-uniformity on $L^X$ is studied. In each case, the relation between their interior operators is obtained. Finally, the relation between the category $\bf Qunif(L,M)$ of all $(L, M)$−fuzzy quasi uniformity spaces and all quasiuniformly continuous functions, and the category $\bf FQunif(P^∗(L),M)$ of all $(P^∗ (L), M)$−fuzzy quasi uniformity spaces and quasi-uniformly continuous fuzzy functions is outlined. It is remarked that all kinds of categories of quasi-uniform spaces and quasi-uniformly continuous functions can be derived from the category $\bf FQunif(P^∗(L),M)$.
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