Parastrophic invariance of Smarandache quasigroups

Abstract

Every quasigroup $(L,\cdot )$ belongs to a set of 6 quasigroups, called parastrophes denoted by $(L,{\pi }_i)$, $i\in \{1,2,3,4,5,6\}$. It is shown that $(L,{\pi }_i)$ is a Smarandache quasigroup with associative subquasigroup $(S,{\pi }_i)\forall i\in \{1,2,3,4,5,6\}$ if and only if for any of some four $j\in \{1,2,3,4,5,6\}$, $(S,{\pi }_j)$ is an isotope of $(S,{\pi }_i)$ or $(S,{\pi }_k)$ for one $k\in \{1,2,3,4,5,6\}$ such that $i\ne j\ne k$. Hence, $(L,{\pi }_i)$ is a Smarandache quasigroup with associative subquasigroup $(S,{\pi }_i)\forall i\in \{1,2,3,4,5,6\}$ if and only if any of the six Khalil conditions is true for any of some four of $(S,{\pi }_i)$.

8 pages