*(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 ${{\Gamma}}^{*}\left(R\right)$ is constructed on $X$ for every fuzzy preordered set $(X,R)$, where ${{\Gamma}}^{*}\left(R\right)$ is the Alexandrov topology generated by $R$. On the other hand, for every fuzzy topological space $(X,\tau )$, a fuzzy preorder ${{\Omega}}^{*}\left(\tau \right)$ on $X$ is constructed; ${{\Omega}}^{*}\left(\tau \right)$ 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.

##### MSC:

54A40 | Fuzzy topology |