# Solving linear equations by fuzzy quasigroups techniques

## TL;DR

This paper introduces a fuzzy quasigroup framework for solving classical linear equations, utilizing lattice-valued fuzzy techniques to analyze solutions and their uniqueness within fuzzy algebraic structures.

## Contribution

It characterizes fuzzy quasigroups via quotient structures and proves the equivalence of two fuzzy quasigroup approaches, extending classical algebraic concepts into fuzzy settings.

## Key findings

- Unique solutions exist for linear equations in fuzzy quasigroups.
- Fuzzy loops with semigroup properties are equivalent to fuzzy groups.
- Procedures for solving linear equations using fuzzy quasigroup properties.

## Abstract

We deal with solutions of classical linear equations ax=b and ya=b, applying a particular lattice valued fuzzy technique. Our framework is a structure with a binary operation (a groupoid), equipped with a fuzzy equality. We call it a fuzzy quasigroup if the above equations have unique solutons with respect to the fuzzy equality. We prove that a fuzzy quasigroup can equivalently be characterized as a structure whose quotients of cut-substructures with respect to cuts of the fuzzy equality are classical quasigroups. Analyzing two approaches to quasigroups in a fuzzy framework, we prove their equivalence. In addition, we prove that a fuzzy loop (quasigroup with a unit element) which is a fuzzy semigroup is a fuzzy group and vice versa. Finally, using properties of these fuzzy quasigroups, we give answers to existence of solutions of the mentioned linear equations with respect to a fuzzy equality, and we describe solving procedures.