Residuated lattices of block relations: size reduction of concept lattices

We deal with size reduction of concept lattices by means of factorization by block relations defined on corresponding formal context. We show that all block relations with a multiplication defined by means of relational composition form a (non-commutative) residuated lattice. Such residuated lattice can be then thought of as a scale of truth degrees using which we evaluate formulas of predicate logic specifying the desired parameters of the factorization. We also introduce efficient algorithms computing operations on a residuated lattice of block relations. The naive way how to design such algorithms is to compute all the formal concepts of a given context in advance, and then apply some well-known properties of residuated lattices. Our algorithms get rid of the time-consuming precomputation of all concepts.