Parastrophic invariance of Smarandache quasigroups

TL;DR

This paper investigates the conditions under which all parastrophes of a quasigroup are Smarandache quasigroups with associative subquasigroups, establishing equivalences involving isotopes and Khalil conditions.

Contribution

It provides a characterization of parastrophic invariance of Smarandache quasigroups using isotopy and Khalil conditions, extending understanding of their algebraic structure.

Findings

All parastrophes are Smarandache quasigroups with associative subquasigroups under certain isotopy conditions.

The equivalence of parastrophic invariance is established via Khalil conditions.

The paper links algebraic invariance to specific isotopic and Khalil condition criteria.

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 $(S,\pi_i)$.