## On linear algebraic semigroups

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- by Mohan S. Putcha PDF
- Trans. Amer. Math. Soc.
**259**(1980), 457-469 Request permission

## 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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## Additional Information

- © Copyright 1980 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**259**(1980), 457-469 - MSC: Primary 20M10
- DOI: https://doi.org/10.1090/S0002-9947-1980-0567091-0
- MathSciNet review: 567091