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On induced $L$−fuzzy uniformities
K. A. DiB,G. A. Kamel,H. M. Rezk 원광대학교 기초자연과학연구소 2017 ANNALS OF FUZZY MATHEMATICS AND INFORMATICS Vol.14 No.2
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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