Fuzzy preorder and fuzzy topology (Q853374)

The paper deals mainly with fuzzy preorder; to be more specific, the categorical aspects of the interrelationship between fuzzy preorder, topological spaces, and fuzzy topological spaces is investigated. The authors delineate basic properties of continuous t-norms and concrete adjoint functors at the beginning; with a brief review on the connection between topological spaces and preordered sets in section 2, section 3 gives a systematic investigation of the properties of upper sets and preordered sets along with several examples. Finally, a fuzzy topology \(\Gamma^*(R)\) is constructed on \(X\) for every fuzzy preordered set \((X,R)\), where \(\Gamma^*(R)\) is the Alexandrov topology generated by \(R\). On the other hand, for every fuzzy topological space \((X,\tau)\), a fuzzy preorder \(\Omega^*(\tau)\) on \(X\) is constructed; \(\Omega^*(\tau)\) is the specialization order on \((X,\tau)\). It is shown that these two constructions are functorial and compatible with their classical counterparts and that the functors \(\Gamma^*\) and \(\Omega^*\) form a pair of adjoint functors between the category of fuzzy preordered sets and that of fuzzy topological spaces.