On linear algebraic semigroups III

Abstract

Using some results on linear algebraic groups, we show that every connected linear algebraic semigroup S contains a closed, connected diagonalizable subsemigroup T with zero such that E ( T ) intersects each regular J ‐class of S . It is also shown that the lattice ( E ( T ), ≤) is isomorphic to the lattice of faces of a rational polytope in some ℝ n . Using these results, it is shown that if S is any connected semigroup with lattice of regular J ‐classes U ( S ), then all maximal chains in U ( S ) have the same length.