On linear algebraic semigroups. II
Mohan S. Putcha · Transactions of the American Mathematical Society · 1980
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