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 is constructed on for every fuzzy preordered set , where is the Alexandrov topology generated by . On the other hand, for every fuzzy topological space , a fuzzy preorder on is constructed; is the specialization order on . It is shown that these two constructions are functorial and compatible with their classical counterparts and that the functors and form a pair of adjoint functors between the category of fuzzy preordered sets and that of fuzzy topological spaces.