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The classification of \(({\mathcal J},\sigma)\)-irreducible monoids of type \(A_4\). (English) Zbl 0907.20052
Assume that \(M\) is a reductive monoid with group of units \(G\). Let \(T\) be a maximal torus of \(G\) and let \(W\) denote the Weyl group of \(G\) relative to \(T\). We say that \(M\) is \((\mathcal J,\sigma)\)-irreducible of type \(A_4\) if \(\sigma\) is an endomorphism of \(M\) such that \((W,\sigma)\) acts transitively on the set of all minimal non-zero idempotents of \(E(\overline T)\) and \(W\) is of type \(A_4\). Then there exist four classes of \((\mathcal J,\sigma)\)-irreducible monoids \(M(\mu)\) of type \(A_4\) where \(\mu\) is an arbitrary dominant weight. The list of cross-section lattices of these monoids is given. Hence there is no general theorem determining the cross-section lattices for \((\mathcal J,\sigma)\)-irreducible monoids of type \(A_4\) according to their Dynkin diagrams. Evidently there is no general theorem for \((\mathcal J,\sigma)\)-irreducible monoids of type \(A_n\), \(n\geq 4\).
Reviewer: V.Koubek (Praha)

20M20 Semigroups of transformations, relations, partitions, etc.
20G15 Linear algebraic groups over arbitrary fields
20M15 Mappings of semigroups
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