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Lattices of fixed points of fuzzy Galois connections. (English) Zbl 0976.03025

The paper is a contribution to the theory of Galois connections, namely the generalization to a residuated lattice \(L\). An \(L\)-Galois connection is a pair of mappings \(\uparrow: L^X\to L^Y\), \(\downarrow: L^Y\to L^X\) satisfying conditions naturally generalizing those for classical Galois connections.
The fundamental theorem states that given a binary \(L\)-relation \(I\in L^{X\times Y}\) in some universes \(X\) and \(Y\) and \(A\in L^X\), \(B\in L^Y\), the following mappings establish an \(L\)-Galois connection: \[ A^{\uparrow}(y)= \bigwedge_{x\in X}(A(x)\rightarrow I(x, y)), \qquad y\in Y, \]
\[ B^{\downarrow}(x)= \bigwedge_{y\in Y}(B(y)\rightarrow I(x, y)), \qquad x\in X. \] A pair \(\langle A, B \rangle\in L^X\times L^Y\) satisfying \(A^{\uparrow}=B\), \(B^{\downarrow}=A\) is a fixed point. In the paper, two theorems are proved, namely: a) \(\langle A, B \rangle\) is a fixed point of the \(L\)-Galois connection \(\langle\uparrow, \downarrow\rangle\) iff it is a maximal rectangle contained in \(I\in L^{X\times Y}\). b) The set \[ \mathbf{L}(X, Y, I)= \{\langle A, B \rangle\in L^X\times L^Y\mid A^{\uparrow}=B, B^{\downarrow}=A\} \] is a complete lattice.

MSC:

03B52 Fuzzy logic; logic of vagueness
06A15 Galois correspondences, closure operators (in relation to ordered sets)
06B15 Representation theory of lattices
03E72 Theory of fuzzy sets, etc.
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