On cyclic associative Abel-Grassman groupoids

TL;DR

This paper introduces cyclic associative Abel-Grassman groupoids (CA-AG-groupoids), explores their properties, enumerates small cases, and distinguishes them from related subclasses within AG-groupoids.

Contribution

It defines CA-AG-groupoids, provides enumeration up to order 6, introduces a verification test for cyclic associativity, and investigates their relationships with other subclasses.

Findings

CA-AG-groupoids are distinct from AG* and AG**-groupoids.

Enumeration of CA-AG-groupoids was achieved up to order 6.

A test for verifying cyclic associativity in AG-groupoids was developed.

Abstract

A new subclass of AG-groupoids, so called, cyclic associative Abel-Grassman groupoids or CA-AG-groupoid is studied. These have been enumerated up to order $6$. A test for the verification of cyclic associativity for an arbitrary AG-groupoid has been introduced. Various properties of CA-AG-groupoids have been studied. Relationship among CA-AG-groupoids and other subclasses of AG-groupoids is investigated. It is shown that the subclass of CA-AG-groupoid is different from that of the AG{*}-groupoid as well as AG{*}{*}-groupoids.