An analogy of Clifford decomposition theorem for Abel-Grassmann groupoids. (English) Zbl 1291.20062
Summary: Let \(S\) be an inverse AG-groupoid (Abel-Grassmann groupoid) and define a relation \(\gamma\) on \(S\) by \(a\gamma b\) if and only if there exist some positive integers \(n\) and \(m\) such that \(b^m\in (Sa)S\) and \(a^n\in (Sb)S\). We prove that \(S/\gamma\) is a maximal semilattice homomorphic image of \(S\). Thus, every inverse AG-groupoid \(S\) is uniquely expressible as a semilattice \(Y\) of some Archimedean inverse AG-groupoids \(S_\alpha\) (\(\alpha\in Y\)). Our result can be regarded as an analogy of the well known Clifford theorem in semigroups for AG-groupoids.
MSC:
20N02 | Sets with a single binary operation (groupoids) |
20M99 | Semigroups |
08A30 | Subalgebras, congruence relations |
Keywords:
Abel-Grassmann groupoids; left invertive law; medial law; congruences; semilattices of Archimedean inverse AG-groupoidsReferences:
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