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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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Connected algebraic monoids
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by Mohan S. Putcha PDF
Trans. Amer. Math. Soc. 272 (1982), 693-709 Request permission

Abstract:

Let $S$ be a connected algebraic monoid with group of units $G$ and lattice of regular $\mathcal {J}$-classes $\mathcal {U}(S)$. The connection between the solvability of $G$ and the semilattice decomposition of $S$ into archimedean semigroups is further elaborated. If $S$ has a zero and if $\mathcal {U}(S)\le 7$, then it is shown that $G$ is solvable if and only if $\mathcal {U}(S)$ is relatively complemented. If $J\in \mathcal {U}(S)$, then we introduce two basic numbers $\theta (J)$ and $\delta (J)$ and study their properties. Crucial to this process is the theorem that for any indempotent $e$ of $S$, the centralizer of $e$ in $G$ is connected. Connected monoids with central idempotents are also studied. A conjecture about their structure is forwarded. It is pointed out that the maximal connected submonoids of $S$ with central idempotents need not be conjugate. However special maximal connected submonoids with central idempotents are conjugate. If $S$ is regular, then $S$ is a Clifford semigroup if and only if for all $f\in E(S)$, the set $\{ e|e \in E(S), e \geq f\}$ is finite. Finally the maximal semilattice image of any connected monoid is determined.
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Additional Information
  • © Copyright 1982 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 272 (1982), 693-709
  • MSC: Primary 20M10; Secondary 20G99
  • DOI: https://doi.org/10.1090/S0002-9947-1982-0662061-8
  • MathSciNet review: 662061