Lattice-theoretic contexts and their concept lattices via Galois ideals

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.