On linear algebraic semigroups. II

Author:
Mohan S. Putcha

Journal:
Trans. Amer. Math. Soc. **259** (1980), 471-491

MSC:
Primary 20M10

DOI:
https://doi.org/10.1090/S0002-9947-80-99945-6

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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* $\{ e | e \in E(S), e \geqslant f\}$ *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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*Natural structures on rings and semigroups with involution*(to appear).

*Problems*23-28, Semigroup Forum

**5**(1972), 92-94.

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Additional Information

Keywords:
Linear algebraic semigroup,
idempotent,
subgroup,
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Article copyright:
© Copyright 1980
American Mathematical Society