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Zbl 1118.54008
Lai, Hongliang; Zhang, Dexue
Fuzzy preorder and fuzzy topology. (English)
[J] Fuzzy Sets Syst. 157, No. 14, 1865-1885 (2006). ISSN 0165-0114

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.
[M. N. Mukherjee (Calcutta)]

MSC 2000:: *54A40 Fuzzy topology

Keywords: fuzzy preorder; upper set; Alexandrov topology; specialization order; fuzzy topology

Cited in: Zbl 1260.54024 Zbl 1182.54016

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