Parastrophes and Cosets of Soft Quasigroups

Anthony Oyem; Temitope Gbolahan Jaiyeola

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arXiv · 2207.06582

Parastrophes and Cosets of Soft Quasigroups

Anthony Oyem·Temitope Gbolahan Jaiyeola

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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.

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BibTeX RIS

Anthony Oyem, Temitope Gbolahan Jaiyeola. 2022-07-14. Parastrophes and Cosets of Soft Quasigroups. https://arxiv.org/abs/2207.06582

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