Measuring and aggregating {\epsilon}-T-transitive fuzzy relations

TL;DR

This paper introduces the concept of { extbackslash epsilon}-T-transitive fuzzy relations, explores measures of T-transitivity, and applies these to clustering and inference tasks, offering a practical approach to handle approximate transitivity.

Contribution

It proposes the { extbackslash epsilon}-T-transitive fuzzy relation concept, characterizes aggregation functions that preserve this property, and demonstrates its application in clustering and inference.

Findings

Two measures of T-transitivity are investigated.

The relationship between different degrees of transitivity is analyzed.

{ extbackslash epsilon}-T-transitive fuzzy relations are effective for clustering with permissible error.

Abstract

The transitivity of fuzzy relations plays an important role in fuzzy set theory, artificial intelligence, clustering and decision-making. However, it is often difficult for fuzzy relations to satisfy the transitivity property in many practical applications. This has motivated researchers to investigate the degree to which a fuzzy relation is transitive. Therefore, this work first investigates two different measures of T-transitivity for fuzzy relations using some well-known fuzzy implications. And then, the relationship between two different degrees of transitivity is investigated. Further, the concept of an {\epsilon}-T-transitive fuzzy relation is introduced, and the aggregation functions that preserve the {\epsilon}-T-transitivity of fuzzy relations are characterized. Finally, the {\epsilon}-T-transitive fuzzy relation is utilized to make inferences and cluster objects. Compared to finding the T-transitive closure, it is reasonable to cluster objects using the {\epsilon}-T-transitive fuzzy relation under the permissible error.