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

Abstract

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 $k\in \{1,2,3,4,5,6\}$.