Chebyshev distances associated to the second members of systems of Max-product/Lukasiewicz Fuzzy relational equations

TL;DR

This paper derives explicit formulas for Chebyshev distances related to the second members of systems of max-product and max-Lukasiewicz fuzzy relational equations, extending previous work on max-min systems.

Contribution

It provides analytical formulas for Chebyshev distances in max-product and max-Lukasiewicz fuzzy relational systems, complementing existing results for max-min systems.

Findings

Formulas for Chebyshev distances in max-product systems

Formulas for Chebyshev distances in max-Lukasiewicz systems

Extension of previous max-min system results

Abstract

In this article, we study the inconsistency of a system of $\max$-product fuzzy relational equations and of a system of $\max$-Lukasiewicz fuzzy relational equations. For a system of $\max-\min$ fuzzy relational equations $A \Box_{\min}^{\max} x = b$ and using the $L_\infty$ norm, (Baaj, 2023) showed that the Chebyshev distance $\Delta = \inf_{c \in \mathcal{C}} \Vert b - c \Vert$, where $\mathcal{C}$ is the set of second members of consistent systems defined with the same matrix $A$, can be computed by an explicit analytical formula according to the components of the matrix $A$ and its second member $b$. In this article, we give analytical formulas analogous to that of (Baaj, 2023) to compute the Chebyshev distance associated to the second member of a system of $\max$-product fuzzy relational equations and that associated to the second member of a system of $\max$-Lukasiewicz fuzzy relational equations.