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 SM ISO690:2012DUDEK, 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 |
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| Volumul 20, Numărul 2(28) / 2012 / ISSN 1561-2848 | ||||||
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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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<?xml version='1.0' encoding='utf-8'?> <oai_dc:dc xmlns:dc='http://purl.org/dc/elements/1.1/' xmlns:oai_dc='http://www.openarchives.org/OAI/2.0/oai_dc/' xmlns:xsi='http://www.w3.org/2001/XMLSchema-instance' xsi:schemaLocation='http://www.openarchives.org/OAI/2.0/oai_dc/ http://www.openarchives.org/OAI/2.0/oai_dc.xsd'> <dc:creator>Dudek, W.A.</dc:creator> <dc:creator>Gigon, R.S.</dc:creator> <dc:date>2012-07-02</dc:date> <dc:description xml:lang='en'>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.</dc:description> <dc:source>Quasigroups and Related Systems 28 (2) 203-209</dc:source> <dc:title>Congruences on completely inverse AG-groupoids</dc:title> <dc:type>info:eu-repo/semantics/article</dc:type> </oai_dc:dc>
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