Induced proximity in fuzzy spaces (Q2793824)

In this paper the authors introduce and study the induced proximity on a fuzzy space from a fuzzy proximity on another fuzzy space. First, for complete lattices \(L\), \(M\) and sets \(Y\subset X\), the extension of an \((L,M)\)-proximity from the fuzzy space \(L^Y\) to \(L^X\) and the restriction of an \((L,M)\)-proximity on \(L^X\) to \(L^Y\) are defined and studied. Second, with \(P(\Lambda)\) denoting the powerset of a nonempty set \(\Lambda\) and \(\mathcal{L}(\Lambda)\) the family of complete lattices on \(\Lambda\), it is shown that every \((L,M)\)-basic proximity \(\delta\) on \(L^X\) (\(L\in\mathcal{L}(\Lambda)\)) induces an \((L,M)\)-basic proximity \(\delta^*\) on \(P(\Lambda)^X\). Finally, the authors show that the category of \((L,M)\)-fuzzy basic proximity spaces on \(L^X\) (\(L\in\mathcal{L}(\Lambda)\)) is embedded in the category of \((P(\Lambda),M)\)-fuzzy basic proximity spaces on \(P(\Lambda)^X\).