On linear algebraic semigroups III

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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 ℝn. 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.

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