# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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by Mohan S. Putcha
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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