Some necessary and sufficient conditions for parastrophic invariance of the associative law in quasigroups

Every quasigroup $(S,\cdot )$ belongs to a set of 6 quasigroups, called parastrophes denoted by $(S,{\pi }_i)$, $i\in \{1,2,3,4,5,6\}$. It is shown that isotopy-isomorphy is a necessary and sufficient condition for any two distinct quasigroups $(S,{\pi }_i)$ and $(S,{\pi }_j)$, $i,j\in \{1,2,3,4,5,6\}$ to be parastrophic invariant relative to the associative law. In addition, a necessary and sufficient condition for any two distinct quasigroups $(S,{\pi }_i)$ and $(S,{\pi }_j)$, $i,j\in \{1,2,3,4,5,6\}$ to be parastrophic invariance under the associative law is either if the ${\pi }_i$-parastrophe of $H$ is equivalent to the ${\pi }_i$-parastrophe of the holomorph of the ${\pi }_i$-parastrophe of $S$ or if the ${\pi }_i$-parastrophe of $H$ is equivalent to the ${\pi }_k$-parastrophe of the ${\pi }_i$-parastrophe of the holomorph of the ${\pi }_i$-parastrophe of $S$, for a particular $k\in \{1,2,3,4,5,6\}$.