## 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 $$\Gamma^*(R)$$ 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.

### MSC:

 54A40 Fuzzy topology
Full Text:

### References:

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