Linear Equations in the Ring of $\mathcal{S}(\mathcal{A})$-Linearly Correlated Fuzzy Numbers

TL;DR

This paper studies solutions to linear fuzzy arithmetic equations involving a special class of fuzzy numbers within a finite-dimensional vector space, utilizing ring operations to analyze their structure.

Contribution

It introduces the concept of $ ext{S}( ext{A})$-linearly correlated fuzzy numbers and explores their role in solving linear equations in a ring structure.

Findings

Characterization of solutions in the ring of $ ext{S}( ext{A})$-linearly correlated fuzzy numbers

Development of algebraic framework for fuzzy linear equations

Extension of fuzzy arithmetic with new operations

Abstract

This paper investigates the solutions of a family of certain linear fuzzy arithmetic equations that involve fuzzy numbers belonging to certain finite-dimensional vector spaces of $\mathbb{R}_{\mathcal{F}}$, called $\mathcal{S}\left(\mathcal{A}\right)$-linearly correlated fuzzy numbers. Here, $\mathcal{A}$ stands for a strongly linearly independent (SLI) set of fuzzy numbers. The arithmetic operations in the aforementioned linear equations are the sum in the vector space $\mathcal{S}(\mathcal{A})$ and the so-called $\psi$-cross product that turns $\mathcal{S}(\mathcal{A})$ into a commutative ring.