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A completely inverse AG∗∗-groupoid is a groupoid satisfying the identities (xy)z=(zy)x, x(yz)=y(xz) and xx−1=x−1x, where x−1 is a unique inverse of x, that is, x=(xx−1)x and x−1=(x−1x)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,...
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