Congruences on completely inverse $A{G}^{\ast \ast }$-groupoids

Abstract

By a completely inverse $A{G}^{\ast \ast }$-groupoid we mean an inverse $A{G}^{\ast \ast }$-groupoid $A$ satisfying the identity $x{x}^{-1}=x^{-1}x$, where $x^{-1}$ denotes a unique element of $A$ such that $x=(x{x}^{-1})x$ and $x^{-1}=(x^{-1}x)x^{-1}.$ We show that the set of all idempotents of such groupoid forms a semilattice and the Green's relations $\mathcal{H},\mathcal{L},\mathcal{R},\mathcal{D}$ and $\mathcal{J}$ coincide on $A$. The main result of this note says that any completely inverse $A{G}^{\ast \ast }$-groupoid meets the famous Lallement's Lemma for regular semigroups. Finally, we show that the Green's relation $\mathcal{H}$ is both the least semilattice congruence and the maximum idempotent-separating congruence on any completely inverse $A{G}^{\ast \ast }$-groupoid.