TY - JOUR AU - Anthony Oyem AU - Tèmítópé Gbóláhàn Jaiyéolá PY - 2022/06/10 Y2 - 2024/03/28 TI - Parastrophes and Cosets of Soft Quasigroups JF - International Journal of Mathematical Sciences and Optimization: Theory and Applications JA - IJMSO VL - 8 IS - 1 SE - Articles DO - 10.6084/m9.figshare.20767495 UR - http://ijmso.unilag.edu.ng/article/view/1584 AB - 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. ER -