Fuzzy closure operators II: induced relations, representation, and examples

Abstract

Closure operators (and related structures) are investigated from the point of view of fuzzy set theory. The paper is a follow up to [7] where fundamental notions and result have been established. The present approach generalizes the existing approaches in two ways: first, complete residuated lattices are used as the structures of truth values (leaving the unite interval [0,1] with minimum and other t-norms particular cases); second, the monotony condition is formulated so that it can reflect also partial subsethood (not only full subsethood as in other approaches). In this paper, we study relations induced by fuzzy closure operators (fuzzy quasiorders and similarities); factorization of closure systems by similarities and by so-called decrease of logical precision; representation of fuzzy closure operators by (crisp) closure operators; relation to consequence relations; and natural examples illustrating the notions and results.