Congruences on completely inverse AG-groupoids

Dudek Wieslaw; Gigon Roman

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DUDEK, Wieslaw, GIGON, Roman. Congruences on completely inverse AG-groupoids. In: Quasigroups and Related Systems, 2012, vol. 20, nr. 2(28), pp. 203-209. ISSN 1561-2848.

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Quasigroups and Related Systems

Volumul 20, Numărul 2(28) / 2012 / ISSN 1561-2848

Congruences on completely inverse AG-groupoids

Pag. 203-209

Dudek Wieslaw, Gigon Roman

Institute of Mathematics and Computer Science, Wroclaw University of Technology

Disponibil în IBN: 25 februarie 2014

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By a completely inverse AG-groupoid we mean an inverse AG-groupoid A satisfying the identity xx−1 = x−1x, where x−1 denotes a unique element of A such that x = (xx−1)x and x−1 = (x−1x)x−1. 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 relationH is both the least semilattice congruence and the maximum idempotent-separating congruence on any completely inverse AG-groupoid.

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