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

TL;DR

This paper characterizes the conditions under which bipolar max-product fuzzy relation equations with standard negation are solvable and explores properties of their solutions, including extremal solutions.

Contribution

It provides a new characterization of solvability for bipolar fuzzy relation equations involving standard negation, along with properties of solutions.

Findings

Conditions for solvability are established.

Properties of extremal solutions are analyzed.

Examples illustrate the theoretical results.

Abstract

Bipolar fuzzy relation equations arise when unknown variables together with their logical negations appear simultaneously in fuzzy relation equations. This paper gives a characterization of the solvability of bipolar max product fuzzy (relation) equations with the standard negation. In addition, some properties associated with the existence of the greatest/least solution or maximal/minimal solutions are shown, when these (relation) equations are solvable. Different examples are included in order to clarify the developed theory.