L-Fuzzy Similarity and L-Fuzzy Distance

Title: L-Fuzzy Similarity and L-Fuzzy Distance

Abstract: This research introduces an L-fuzzy valued inclusion measure, similarity measure, and distance function between fuzzy sets. These measures possess properties analogous to real-valued similarity and distance functions. The main difference between this work and previous research is the use of a Boolean lattice as the range for the measures, making them L-fuzzy valued relations between fuzzy sets. These measures are reflexive, symmetric, and transitive, and they satisfy a number of properties postulated as axiomatically appropriate for an inclusion measure.

Research Question: Can we develop L-fuzzy valued relations between fuzzy sets that serve as inclusion, similarity, and distance measures, and do these measures possess desirable properties?

Methodology: The study begins by introducing an L-fuzzy valued inclusion measure, denoted as I(A, B), between fuzzy sets A and B. This measure is reflexive, symmetric, and transitive, making it an L-fuzzy order relation. The paper also introduces an L-fuzzy valued similarity measure, denoted as S(A, B), and a distance function, D(A, B). All measures are defined using a Boolean lattice, which allows them to be vector-valued instead of scalar-valued.

Results: The proposed inclusion measure, similarity measure, and distance function all possess properties analogous to their real-valued counterparts. Additionally, the measures are shown to be reflexive, symmetric, and transitive, making them L-fuzzy order relations.

Implications: The use of a Boolean lattice as the range for the measures allows them to be vector-valued, which provides more flexibility and nuance than scalar-valued measures. This can lead to more accurate and meaningful results in various applications. The proposed measures also satisfy a number of properties postulated as axiomatically appropriate for an inclusion measure, which further highlights their utility and potential for broad application.

Link to Article: https://arxiv.org/abs/0111002v1 Authors: arXiv ID: 0111002v1