IMCS/Publications/QRS/Issues/QRS v.20, n.2 (28), 2012/

Congruences on completely inverse AG**-groupoids

Authors: W. A. Dudek, R. S. Gigon


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


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