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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On linear algebraic semigroups


Author: Mohan S. Putcha
Journal: Trans. Amer. Math. Soc. 259 (1980), 457-469
MSC: Primary 20M10
MathSciNet review: 567091
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Abstract: Let K be an algebraically closed field. By an algebraic semigroup we mean a Zariski closed subset of $ {K^n}$ along with a polynomially defined associative operation. Let S be an algebraic semigroup. We show that S has ideals $ {I_0},\, \ldots \,,\,{I_t}$ such that $ S\, = \,{I_t}\, \supseteq \, \cdots \, \supseteq \,{I_0}$, $ {I_0}$ is the completely simple kernel of S and each Rees factor semigroup $ {I_k}/{I_{k - 1}}$ is either nil or completely 0-simple $ (k\, = \,1,\, \ldots \,,\,t)$. We say that S is connected if the underlying set is irreducible. We prove the following theorems (among others) for a connected algebraic semigroup S with idempotent set $ E(S)$. (1) If $ E(S)$ is a subsemigroup, then S is a semilattice of nil extensions of rectangular groups. (2) If all the subgroups of S are abelian and if for all $ a\, \in \,S$, there exists $ e\, \in \,E(S)$ such that $ ea\, = \,ae\, = a$, then S is a semilattice of nil extensions of completely simple semigroups. (3) If all subgroups of S are abelian and if S is regular, then S is a subdirect product of completely simple and completely 0-simple semigroups. (4) S has only trivial subgroups if and only if S is a nil extension of a rectangular band.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1980-0567091-0
PII: S 0002-9947(1980)0567091-0
Keywords: Linear algebraic semigroup, idempotent, subgroup, $ \mathcal{J}$-class
Article copyright: © Copyright 1980 American Mathematical Society