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We deal with solutions of classical linear equations ax=b and ya=b, applyinga particular lattice valued fuzzy technique. Our framework is a structure witha binary operation (a groupoid), equipped with a fuzzy equality. We call it afuzzy quasigroup if the above equations have unique solutons with respect tothe fuzzy equality. We prove that a fuzzy quasigroup can equivalently becharacterized as a structure whose quotients of cut-substructures with respectto cuts of the fuzzy equality are classical quasigroups. Analyzing twoapproaches to quasigroups in a fuzzy framework, we prove their equivalence. Inaddition, we prove that a fuzzy loop (quasigroup with a unit element) which isa fuzzy semigroup is a fuzzy group and vice versa. Finally, using properties ofthese fuzzy quasigroups, we give answers to existence of solutions of thementioned linear equations with respect to a fuzzy equality, and we describesolving procedures.

Solving linear equations by fuzzy quasigroups techniques