  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.
