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We study the geometry of algebraic monoids. We prove that the group of invertible elements of an irreducible algebraic monoid is an algebraic group, open in the monoid. Moreover, if this group is reductive, then the monoid is affine. We then give a combinatorial classification of reductive monoids by means of the theory of spherical varieties.
Let G be an affine algebraic group and let X be an affine algebraic variety. An action G × X → X is called observable if for any G-invariant, proper, closed subset Y of X there is a nonzero invariant f ∈ k[X] G such that f | Y = 0. We characterize this condition geometrically as follows. The action G × X → X is observable if and only if (1) there is a(More)
This paper is intended to very briefly introduce some computation problems in linear algebraic monoids. The theory of linear algebraic monoids was initiated independently in 1980 Ontario). This theory is a natural blend of algebraic groups, torus embeddings, and semigroups. It is a very active and fruitful research area in mathematics. Many other(More)
A closed subgroup H of the affine, algebraic group G is called observable if G/H is a quasi-affine algebraic variety. In this paper we define the notion of an observable subgroup of the affine, algebraic monoid M. We prove that a subgroup H of G is observable in M if and only if H is closed in M and there are " enough " H-semiinvariant functions in k[M]. We(More)
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