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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
DOI: https://doi.org/10.1090/S0002-9947-1982-0662061-8
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\vert e \in E(S),\,e \geq f\} $ is finite. Finally the maximal semilattice image of any connected monoid is determined.


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DOI: https://doi.org/10.1090/S0002-9947-1982-0662061-8
Keywords: Connected monoids, solvable groups, $ \mathcal{J}$-class, lattice, semilattice, idempotents
Article copyright: © Copyright 1982 American Mathematical Society