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 U ( S ) \mathcal {U}(S) be the partially ordered set of regular J \mathcal {J} -classes of S and let E ( S ) E(S) be the set of idempotents of S. The following theorems (among others) are proved. (1) U ( S ) \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 ) E(S) . (3) If S is a Clifford semigroup and f ∈ E ( S ) f\, \in \,E(S) , then the set { e | e ∈ E ( S ) , e ⩾ f } \{ \,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 J \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 , ⋅ ) ({K^n},\, \cdot ) for some n ∈ Z + n\, \in \,{\textbf {Z}^ + } . (6) If ch ⋅ K = 0 {\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 , ⋅ ) ({K^n},\, \cdot ) for some n ∈ Z + n\, \in \,{\textbf {Z}^ + } . (7) If S is a regular semigroup and dim ⋅ S ⩽ 2 {\text {dim}}\, \cdot \,S\, \leqslant \,2 , then | U ( S ) | ⩽ 4 \left | {\mathcal {U}(S)} \right |\, \leqslant \,4 . (8) If S is a Clifford semigroup with zero and dim ⋅ S = 3 {\text {dim}}\, \cdot \,S\, = \,3 , then | E ( S ) | = | U ( S ) | \left | {E(S)} \right |\, = \,\left | {\mathcal {U}(S)} \right | can be any even number ⩾ 8 \geqslant \,8 . (9) If S is a Clifford semigroup then U ( S ) \mathcal {U}(S) is a relatively complemented lattice and all maximal chains in U ( S ) \mathcal {U}(S) have the same number of elements.