Authors: Krapež, Aleksandar 
Šešelja, Branimir
Tepavčević, Andreja 
Title: Solving linear equations by fuzzy quasigroups techniques
Journal: Information Sciences
Volume: 491
First page: 179
Last page: 189
Issue Date: 1-Jul-2019
Rank: M21a
ISSN: 0020-0255
DOI: 10.1016/j.ins.2019.03.073
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.
Keywords: Fuzzy equality | Lattice valued | Linear equation | L–quasigroup
Publisher: Elsevier
Project: Advanced Techniques of Cryptology, Image Processing and Computational Topology for Information Security 
Representations of logical structures and formal languages and their application in computing 
Development of methods of computation and information processing: theory and applications 

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