Right product quasigroups and loops

TL;DR

This paper introduces and axiomatizes right product quasigroups and loops, extending the concept of right groups to non-associative structures and providing foundational identities and properties.

Contribution

It defines right product quasigroups and loops, establishes their axioms, and explores their properties, extending the theory of right groups to non-associative contexts.

Findings

A system of identities for right product quasigroups is established.

Axioms for right product loops are derived by adding one axiom.

The axioms for right product quasigroups are shown to be independent.

Abstract

Right groups are direct products of right zero semigroups and groups and they play a significant role in the semilattice decomposition theory of semigroups. Right groups can be characterized as associative right quasigroups (magmas in which left translations are bijective). If we do not assume associativity we get right quasigroups which are not necessarily representable as direct products of right zero semigroups and quasigroups. To obtain such a representation, we need stronger assumptions which lead us to the notion of \emph{right product quasigroup}. If the quasigroup component is a (one-sided) loop, then we have a \emph{right product (left, right) loop}. We find a system of identities which axiomatizes right product quasigroups, and use this to find axiom systems for right product (left, right) loops; in fact, we can obtain each of the latter by adjoining just one appropriate axiom to the right product quasigroup axiom system. We derive other properties of right product quasigroups and loops, and conclude by showing that the axioms for right product quasigroups are independent.