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Transactions of the American Mathematical Society

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



Connected algebraic monoids

Author: Mohan S. Putcha
Journal: Trans. Amer. Math. Soc. 272 (1982), 693-709
MSC: Primary 20M10; Secondary 20G99
MathSciNet review: 662061
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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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Keywords: Connected monoids, solvable groups, <!– MATH $\mathcal {J}$ –> <IMG WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="$\mathcal {J}$">-class, lattice, semilattice, idempotents
Article copyright: © Copyright 1982 American Mathematical Society