Congruences on completely inverse $AG^{**}$-groupoids

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