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arXiv · 0706.1480

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

Temitope Gbolahan Jaiyeola
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Abstract

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

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Temitope Gbolahan Jaiyeola. 2009-03-10. Some necessary and sufficient conditions for parastrophic invariance of the associative law in quasigroups. https://arxiv.org/abs/0706.1480

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