Completely inverse $AG^{**}$-groupoids

TL;DR

This paper explores the properties, fundamental congruences, and the lattice structure of completely inverse $AG^{**}$-groupoids, providing a detailed algebraic framework for understanding their structure and classifications.

Contribution

It introduces and analyzes the structure of completely inverse $AG^{**}$-groupoids, including their key congruences and lattice of all congruences, which was not previously studied.

Findings

Determined fundamental properties of completely inverse $AG^{**}$-groupoids.

Identified key congruences: maximum idempotent-separating, least $AG$-group, and least $E$-unitary.

Described the lattice of congruences via kernels and traces.

Abstract

A completely inverse $AG^{**}$-groupoid is a groupoid satisfying the identities $(xy)z=(zy)x$, $x(yz)=y(xz)$ and $xx^{-1}=x^{-1}x$, where $x^{-1}$ is a unique inverse of $x$, that is, $x=(xx^{-1})x$ and $x^{-1}=(x^{-1}x)x^{-1}$. First we study some fundamental properties of such groupoids. Then we determine certain fundamental congruences on a completely inverse $AG^{**}$-groupoid; namely: the maximum idempotent-separating congruence, the least $AG$-group congruence and the least $E$-unitary congruence. Finally, we investigate the complete lattice of congruences of a completely inverse $AG^{**}$-groupoids. In particular, we describe congruences on completely inverse $AG^{**}$-groupoids by their kernel and trace.