L-Fuzzy Valued Inclusion Measure, L-Fuzzy Similarity and L-Fuzzy Distance

Abstract

The starting point of this paper is the introduction of a new measure of inclusion of fuzzy set A in fuzzy set B. Previously used inclusion measures take values in the interval [0,1]; the inclusion measure proposed here takes values in a Boolean lattice. In other words, inclusion is viewed as an L-fuzzy valued relation between fuzzy sets. This relation is re exive, antisymmetric and transitive, i.e. it is a fuzzy order relation; in addition it possesess a number of properties which various authors have postulated as axiomatically appropriate for an inclusion measure. We also define an L-fuzzy valued measure of similarity between fuzzy sets and and an L-fuzzy valued distance function between fuzzy sets; these possess properties analogous to the ones of real-valued similarity and distance functions. Keywords: Fuzzy Relations, inclusion measure, subsethood, L-fuzzy sets, similarity, distance, transitivity.