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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)

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