On linear algebraic semigroups. II

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.