A Pair of Smarandachely Isotopic Quasigroups and Loops of the Same Variety

TL;DR

This paper explores the conditions under which a pair of Smarandache isotopic quasigroups or loops belong to the same variety, addressing open problems in loop theory and providing constructs useful for applications like cryptography.

Contribution

It redefines Smarandache isotopism to explore its role in the universality of quasigroup and loop varieties, solving a specific open problem.

Findings

Presented a pair of S-isotopic S-quasigroups in the same variety.

Connected S-isotopism with open problems in loop theory.

Highlighted applications in cryptography.

Abstract

The isotopic invariance or universality of types and varieties of quasigroups and loops described by one or more equivalent identities has been of interest to researchers in loop theory in the recent past. A variety of quasigroups(loops) that are not universal have been found to be isotopic invariant relative to a special type of isotopism or the other. Presently, there are two outstanding open problems on universality of loops: semi automorphic inverse property loops(1999) and Osborn loops(2005). Smarandache isotopism(S-isotopism) was originally introduced by Vasantha Kandasamy in 2002. But in this work, the concept is re-restructured in order to make it more explorable. As a result of this, the theory of Smarandache isotopy inherits the open problems as highlighted above for isotopy. In this short note, the question 'Under what type of S-isotopism will a pair of S-quasigroups(S-loops) form any variety?' is answered by presenting a pair of specially S-isotopic S-quasigroups(loops) that both belong to the same variety of S-quasigroups(S-loops). This is important because pairs of specially S-isotopic S-quasigroups(e.g Smarandache cross inverse property quasigroups) that are of the same variety are useful for applications(e.g cryptography).