Fuzzy Galois connections. (English) Zbl 0938.03079

Recall that a (contravariant) Galois connection between two posets \(P\) and \(Q\) is a pair of mappings \(P@>+>>Q\), \(Q@>->>P\) such that \(a\leq b^-\Leftrightarrow b\leq a^+\). The standard example is the polarity associated with a relation \(\rho\subseteq X\times Y\), i.e., the Galois connection between \(\mathbf{2}^X\) and \(\mathbf{2}^Y\) defined by \(A^+= \{y\in Y\mid x\rho y\) \(\forall x\in A\}\) and \(B^-= \{x\in X\mid x\rho y\) \(\forall y\in B\}\). Further, let \((L;\wedge,\vee, 0,1,\rightarrow)\) be a complete residuated lattice. The author defines a fuzzy Galois connection between two fuzzy sets \(L^X\) and \(L^Y\) as a pair of mappings \(L^X@>+>> L^Y\), \(L^Y@>->>L^X\) such that \(\text{Subs}(A,B^-)= \text{Subs}(B, A^+)\), where the subsethood degree \(\text{Subs}(A_1,A_2)\) is defined by \(\text{Subs}(A_1,A_2)= \inf\{A_1(x)\to A_2(x)\mid x\in X\}\). Furthermore, the fuzzy polarity associated with a fuzzy relation \(R\in L^{X\times Y}\) is defined by \(A^+(y)= \inf\{A(x)\to R(x,y)\mid x\in X\}\) and \(B^-(x)= \inf\{B(y)\to R(x,y)\mid y\in Y\}\). The main result of the paper is a bijection between fuzzy Galois connections and fuzzy relations, such that every fuzzy Galois connection is the fuzzy polarity determined by the associated fuzzy relation, and every fuzzy relation is associated with a fuzzy Galois connection in the way indicated above. This generalizes a theorem of Ore on Galois connections.
Remark: The author has informed the reviewer that the formula on page 498, line 22, should read \(\{a/x\}(x')= 0\).


03E72 Theory of fuzzy sets, etc.
06A15 Galois correspondences, closure operators (in relation to ordered sets)
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