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 be the partially ordered set of regular -classes of *S* and let be the set of idempotents of *S*. The following theorems (among others) are proved. (1) *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* . (3) *If S is a Clifford semigroup and* , *then the set* *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* -*class of S*. (5) *If S is a Clifford semigroup with zero, then S is commutative and is in fact embeddable in* *for some* . (6) *If* *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* *for some* . (7) *If S is a regular semigroup and* , *then* . (8) *If S is a Clifford semigroup with zero and* , *then* *can be any even number* . (9) *If S is a Clifford semigroup then* *is a relatively complemented lattice and all maximal chains in* *have the same number of elements*.

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Additional Information

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

Keywords:
Linear algebraic semigroup,
idempotent,
subgroup,
-class

Article copyright:
© Copyright 1980
American Mathematical Society