Varieties of groupoids and quasigroups generated by linear-bivariate polynomials over ring Z_n

TL;DR

This paper investigates varieties of groupoids and quasigroups generated by linear-bivariate polynomials over the ring Z_n, establishing conditions for identities and inverse properties, and classifying them into known algebraic varieties.

Contribution

It provides necessary and sufficient conditions for such algebraic structures to satisfy specific identities and inverse properties, expanding understanding of their classification.

Findings

Identifies conditions for identities involving multiple variables

Classifies generated structures into known varieties

Establishes inverse property conditions

Abstract

Some varieties of groupoids and quasigroups generated by linear-bivariate polynomials $P(x,y)=a+bx+cy$ over the ring $\mathbb{Z}_n$ are studied. Necessary and sufficient conditions for such groupoids and quasigroups to obey identities which involve one, two, three (e.g. Bol-Moufang type) and four variables w.r.t. $a$, $b$ and $c$ are established. Necessary and sufficient conditions for such groupoids and quasigroups to obey some inverse properties w.r.t. $a$, $b$ and $c$ are also established. This class of groupoids and quasigroups are found to belong to some varieties of groupoids and quasigroups such as medial groupoid(quasigroup), F-quasigroup, semi automorphic inverse property groupoid(quasigroup) and automorphic inverse property groupoid(quasigroup).