Topology on residuated lattices for biresiduum-based continuity of fuzzy sets

Abstract

By continuity of $[0,1]$-valued fuzzy set it is usually understood standard continuity of a real-valued function. This could be in conflict with the similarity fuzzy relation on $[0,1]$ given by the operation of biresiduum induced by a left-continuous t-norm. In this work, a new kind of continuity, called t-continuity, of an L-set in a topological space for L being a complete residuated lattice of finite dimension is defined. It is based on a notion of oscillation of L-set, which is an element of L computed by means of biresiduum in L. A topology on L of this continuity, called the t-topology on L, is found. This topology is constructed by means of biresiduum and a variant of Scott-open sets called residuated Scott-open sets. It is shown that the t-topology of L is Hausdorff and that L together with its t-topology is a topological residuated lattice. T-topologies of four standard t-norms are found. The notion of t-topology is generalized to sets with L-equivalence relation and a new characteristic of extensional sets implying that extensional L-sets are t-continuous is given.