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Lattice-theoretic contexts and their concept lattices via Galois ideals. (English) Zbl 1395.68258
Summary: This paper introduces a concept of lattice-theoretic contexts as well as their concept lattices. A lattice-theoretic context is a triple $$(G, M, I)$$ with two complete lattices $$G$$, $$M$$ and their Galois ideal $$I$$. A lattice-theoretic context and its concept lattice are a common generalization of classical FCA, Pócs’s formal fuzzy context, one-sided concept lattices, generalized concept lattices and $$L$$-fuzzy concept lattices (with hedges). When the lattices $$G$$, $$M$$ are completely distributive, a reduction of the relation $$I$$ in the lattice-theoretic context $$(G, M, I)$$ can be obtained. Related algorithms to construct concept lattices of $$L$$-fuzzy contexts considered as lattice-theoretic contexts are presented. In the case of $$L$$ being a completely distributive lattice, we can reduce the number of elements (objects or/and attributes) before computing the whole concept lattice. Then the related algorithm has lower complexity.

##### MSC:
 68T30 Knowledge representation 06A15 Galois correspondences, closure operators (in relation to ordered sets) 06D10 Complete distributivity
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