## 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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