Bipolar fuzzy relation equations systems based on the product t-norm

TL;DR

This paper investigates the solvability and algebraic structure of bipolar fuzzy relation equations systems based on the max-product t-norm, extending previous work on their solvability and considering systems with zero independent terms.

Contribution

It provides new insights into the algebraic properties and solvability conditions of bipolar fuzzy relation equations using the max-product t-norm, including special cases with zero independent terms.

Findings

Characterization of the solution set structure

Conditions for solvability of bipolar fuzzy relation systems

Extension of previous results to systems with zero independent terms

Abstract

Bipolar fuzzy relation equations arise as a generalization of fuzzy relation equations considering unknown variables together with their logical connective negations. The occurrence of a variable and the occurrence of its negation simultaneously can give very useful information for certain frameworks where the human reasoning plays a key role. Hence, the resolution of bipolar fuzzy relation equations systems is a research topic of great interest. This paper focuses on the study of bipolar fuzzy relation equations systems based on the max-product t-norm composition. Specifically, the solvability and the algebraic structure of the set of solutions of these bipolar equations systems will be studied, including the case in which such systems are composed of equations whose independent term be equal to zero. As a consequence, this paper complements the contribution carried out by the authors on the solvability of bipolar max-product fuzzy relation equations.