Solving linear equations by fuzzy quasigroups techniques

Abstract

We deal with solutions of classical linear equations a·x=b and y·a=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 solutions 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. Quasigroups and other related structures are an algebraic tool successfully applied up to now in coding theory and cryptology. In our work we propose related applications.