Parastrophes and Cosets of Soft Quasigroups

TL;DR

This paper explores the structure of soft quasigroups, introducing concepts like parastrophes, cosets, and normality, and establishes conditions and properties relating to their distributive, normal, and quotient structures.

Contribution

It introduces the concept of soft quasigroups and their parastrophes, providing new conditions and properties, especially regarding cosets, normality, and distributivity.

Findings

A soft set over a group is a soft group if and only if it is a soft loop or two of its parastrophes are soft groupoids.

In finite quasigroups, the orders of a soft quasigroup and its parastrophes are equal.

Distributive soft quasigroups have all parastrophes distributive, idempotent, and flexible.

Abstract

This paper introduced the concept of soft quasigroup, its parastrophes, soft nuclei, left (right) coset, distributive soft quasigroups and normal soft quasigroups. Necessary and sufficient conditions for a soft set over a quasigroup (loop) to be a soft quasigroup (loop) were established. It was proved that a soft set over a group is a soft group if and only if it is a soft loop or either of two of its parastrophes is a soft groupoid. For a finite quasigroup, it was shown that the orders (arithmetic and geometric means) of the soft quasigroup over it and its parastrophes are equal. It was also proved that if a soft quasigroup is distributive, then all its parastrophes are distributive, idempotent and flexible soft quasigroups. For a distributive soft quasigroup, it was shown that its left and right cosets form families of distributive soft quasigroups that are isomorphic. If in addition, a soft quasigroup is normal, then its left and right cosets forms families of normal soft quasigroups. On another hand, it was found that if a soft quasigroup is a normal and distributive soft quasigroup, then its left (right) quotient is a family of commutative distributive quasigroups which have a 1-1 correspondence with the left (right) coset of the soft quasigroup.