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 $(S,\pi_i)$.