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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