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Fuzzifying syntopogenous structures. (English) Zbl 1070.54004
The authors define a fuzzifying topogeneous order on a set \(X\) to be a map \(\xi: 2^X\times 2^X\to [0,1]\) satisfying the following axioms:
1) \(\xi(X,X)= \xi(\phi,\phi)= 1\);
2) \(\xi(A,B)= 0\) if \(A\) is not contained in \(B\);
3) \(A_1\subset A\subset B\subset B_1\) implies that \(\xi(A_1,B_1)\geq \xi(A,B)\);
4) \(\xi(A_1\cup A_2,B)\geq \xi(A,B)\wedge \xi(A_2,B)\) and \(\xi(A,B_1\cap B_2)\geq \xi(A,B)\wedge\xi(A,B_2)\).
The complement of a fuzzifying topogeneous order \(\xi\) is the fuzzifying topogeneous order \(\xi^c\) defined by \(\xi^c(A,B)= \xi(B^c,A^c)\).
A fuzzifying syntopogeneous structure on a set \(X\) is a nonempty family \(S\) of fuzzifying topogeneous orders on \(X\) such that:
1) \(S\) is directed in the sense that, given \(\xi_1, \xi_2\in S\) there exists \(\xi\in S\) with \(\xi\geq\xi_1,\xi_2\);
2) Given \(\xi\in S\) and \(\varepsilon> 0\) there exists \(\xi_1\in S\) with \(\xi_1\circ\xi_1+ \varepsilon\geq \xi\). If \(S\) consists of a single point, then the structure is called topogeneous. In the process, perfect, biperfect and symmetrical syntopogeneous structures are defined.
Among other results, in the paper, one-to-one correspondences, as follows, are established:
1) between fuzzifying topologies and perfect fuzzifying syntopogeneous structures;
2) fuzzifying proximities and the symmetrical fuzzifying topogeneous structures;
3) fuzzy uniformities and the symmetrical biperfect fuzzifying syntopogeneous structures.

MSC:
54A40 Fuzzy topology
54A15 Syntopogeneous structures
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