On the solvability of bipolar max-product fuzzy relation equations with the product negation

TL;DR

This paper investigates the solvability of bipolar max-product fuzzy relation equations incorporating a residuated negation, expanding the understanding of their solutions across various scenarios.

Contribution

It introduces a residuated negation of the product t-norm into fuzzy relation equations, enhancing their flexibility and analyzing solution sets in different variable and equation configurations.

Findings

Solvability conditions depend on the number of variables and equations.

The set of solutions varies with different scenarios.

Residuated negation increases the flexibility of fuzzy relation equations.

Abstract

This paper studies the solvability of the max-product fuzzy relation equations in which a negation operator is considered. Specifically, the residuated negation of the product t-norm has been introduced in these equations in order to increase the flexibility of the standard fuzzy relation equations introduced by Sanchez in 1976. The solvability and the set of solutions of these bipolar equations have been studied in different scenarios, depending on the considered number of variables and equations.