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Fuzzy preorder and fuzzy topology. (English) Zbl 1118.54008

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

54A40Fuzzy topology