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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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On linear algebraic semigroups. II
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by Mohan S. Putcha PDF
Trans. Amer. Math. Soc. 259 (1980), 471-491 Request permission

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 $\{ e | e \in E(S), e \geqslant f\}$ 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 $\left | {\mathcal {U}(S)} \right | \leqslant 4$. (8) If S is a Clifford semigroup with zero and ${\text {dim}} \cdot S = 3$, then $\left | {E(S)} \right | = \left | {\mathcal {U}(S)} \right |$ 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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Additional Information
  • © Copyright 1980 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 259 (1980), 471-491
  • MSC: Primary 20M10
  • DOI: https://doi.org/10.1090/S0002-9947-80-99945-6