Bipolar fuzzy relation equations systems based on the product t-norm
M. Eugenia Cornejo, David Lobo, Jesús Medina
TL;DR
This work extends bipolar fuzzy relation equations to the product t-norm setting with product negation, addressing both single equations and systems. It provides a complete solvability characterization via feasibility of index-set pairs, and furnishes explicit solution constructions alongside the analysis of the solution-set algebra, including cases with zero independent terms. The authors establish connections between feasible pairs and extremal solutions (greatest/least and maximal/minimal) and illustrate the approach with a practical motor overheating example. The results close an open problem from prior work and lay groundwork for future exploration of arbitrary negations and other t-norm-based operators, enhancing the applicability of bipolar FREs in reasoning-driven frameworks.
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.