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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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##### References:
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