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The lattice of \(\mathcal J\)-classes of \(({\mathcal J},\sigma)\)-irreducible monoids. (English) Zbl 0879.20035
Let \(M\) be a reductive algebraic monoid, that is, a Zariski closed irreducible monoid whose group of units \(G\) is a reductive group. Let \(T\), \(W\) be a maximal torus of \(G\) and the corresponding Weyl group. \(M\) is said to be \(({\mathcal J},\sigma)\)-irreducible if \(\sigma\) is an endomorphism of \(M\) such that \((W,\sigma)\) acts transitively on the set \(E_1(\overline T)\) of minimal nonzero idempotents of \(\overline T\). In other words, for \(e\in E_1(\overline T)\) we have \(E_1(\overline T)=We\cup W\sigma(e)\cup W\sigma^2(e)\cup\cdots\). For example, if \(\sigma=1\) and \(M\) is \(\mathcal J\)-irreducible (that is, all minimal nonzero idempotents are conjugate), then \(M\) is of the above type. This latter special case was studied in particular by M. S. Putcha and the second author [in J. Algebra 116, No. 2, 385-399 (1988; Zbl 0678.20039)]. Starting from a simple algebraic group \(G_0\) and a surjective endomorphism \(\sigma\) of \(G_0\) with finite fixed points \((G_0)_\sigma\), the authors construct a \(({\mathcal J},\sigma)\)-irreducible monoid \(M\) with the unit group \((G_0)_\sigma\). Extending \(\sigma\) to \(M\) they get a finite monoid \(M_\sigma\), which is \(\mathcal J\)-irreducible. The lattices of \(\mathcal J\)-classes of \(M\) and of \(M_\sigma\) are then studied.

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