Powerset operator foundations for poslat fuzzy set theories and topologies (Q2702342)

Given a mapping \(f\) from \(X\) to \(Y\), using the concepts of direct and inverse image it may be extended to a mapping between the powerset \({\mathcal P}(X)\) of \(X\), and the powerset \({\mathcal P}(Y)\) of \(Y\). Zadeh extended these mappings to the class \({\mathcal F}(X)\) of fuzzy sets in \(X\) and the class \({\mathcal F}(Y)\) of fuzzy sets in \(Y\) by the well-known extension principle. In this paper the author extends these concepts to \(L\)-fuzzy sets in \(X\) as domain and \(M\)-fuzzy sets in \(Y\) where \(L\) and \(M\) denote complete quasi-monoidal lattices with a given morphism between them: the author calls this extension the variable-basis (evaluation set in the usual terminology) case. Most results in this paper are extensions of an earlier version [Quaest. Math. 20, 463-530 (1997; Zbl 0911.04003)] and proofs are omitted for reasons of similarity with the ones in the previous paper.NEWLINENEWLINEFor the entire collection see [Zbl 0942.00008].