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Transactions of the American Mathematical Society

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

 

 

On linear algebraic semigroups. II


Author: Mohan S. Putcha
Journal: Trans. Amer. Math. Soc. 259 (1980), 471-491
MSC: Primary 20M10
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Abstract: We continue from [11] the study of linear algebraic semigroups. Let S be a connected algebraic semigroup defined over an algebraically closed field K. Let $ \mathcal{U}(S)$ be the partially ordered set of regular $ \mathcal{J}$-classes of S and let $ E(S)$ be the set of idempotents of S. The following theorems (among others) are proved. (1) $ \mathcal{U}(S)$ is a finite lattice. (2) If S is regular and the kernel of S is a group, then the maximal semilattice image of S is isomorphic to the center of $ E(S)$. (3) If S is a Clifford semigroup and $ f\, \in \,E(S)$, then the set $ \{ \,e\,\vert\,e\, \in \,E(S),\,e\, \geqslant \,f\} $ is finite. (4) If S is a Clifford semigroup, then there is a commutative connected closed Clifford subsemigroup T of S with zero such that T intersects each $ \mathcal{J}$-class of S. (5) If S is a Clifford semigroup with zero, then S is commutative and is in fact embeddable in $ ({K^n},\, \cdot )$ for some $ n\, \in \,{\textbf{Z}^ + }$. (6) If $ {\text{ch}}\, \cdot \,K\, = \,0$ and S is a commutative Clifford semigroup, then S is isomorphic to a direct product of an abelian connected unipotent group and a closed connected subsemigroup of $ ({K^n},\, \cdot )$ for some $ n\, \in \,{\textbf{Z}^ + }$. (7) If S is a regular semigroup and $ {\text{dim}}\, \cdot \,S\, \leqslant \,2$, then $ \left\vert {\mathcal{U}(S)} \right\vert\, \leqslant \,4$. (8) If S is a Clifford semigroup with zero and $ {\text{dim}}\, \cdot \,S\, = \,3$, then $ \left\vert {E(S)} \right\vert\, = \,\left\vert {\mathcal{U}(S)} \right\vert$ can be any even number $ \geqslant \,8$. (9) If S is a Clifford semigroup then $ \mathcal{U}(S)$ is a relatively complemented lattice and all maximal chains in $ \mathcal{U}(S)$ have the same number of elements.


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DOI: https://doi.org/10.1090/S0002-9947-80-99945-6
Keywords: Linear algebraic semigroup, idempotent, subgroup, $ \mathcal{J}$-class
Article copyright: © Copyright 1980 American Mathematical Society