Completely inverse $A{G}^{\ast \ast }$-groupoids

Abstract

A completely inverse $A{G}^{\ast \ast }$-groupoid is a groupoid satisfying the identities $(xy)z=(zy)x$, $x(yz)=y(xz)$ and $x{x}^{-1}=x^{-1}x$, where $x^{-1}$ is a unique inverse of $x$, that is, $x=(x{x}^{-1})x$ 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 $A{G}^{\ast \ast }$-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 $A{G}^{\ast \ast }$-groupoids. In particular, we describe congruences on completely inverse $A{G}^{\ast \ast }$-groupoids by their kernel and trace.

Cited by in corpus (1)

• Intra-regular Abel-Grassmann's groupoids