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arXiv · 1701.07701

Solving linear equations by fuzzy quasigroups techniques

Aleksandar Krapež·Branimir Šešelja·Andreja Tepavčević
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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.

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Aleksandar Krapež, Branimir Šešelja, Andreja Tepavčević. 2017-01-26. Solving linear equations by fuzzy quasigroups techniques. https://arxiv.org/abs/1701.07701

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