Finite full transformation semigroups as collections of random functions

The collection of all self-maps on a non-empty set X under composition is known in algebraic semigroup theory as the full transformation semigroup on X and is written x. Its importance lies in the fact that any semigroup S can be embedded in the full transformation semigroup (where S1 is the semigroup S with identity 1 adjoined, if S does not already possess one). The proof is similar to Cayley's Theorem that a group G can be embedded in SG, the group of all bijections of G to itself. In this paper X will be a finite set of order n, which we take to be and so we shall write Tn for X.

On linear algebraic semigroups III

Using some results on linear algebraic groups, we show that every connected linear algebraic semigroupScontains a closed, connected diagonalizable subsemigroupTwith zero such thatE(T)intersects each regularJ-class ofS. It is also shown that the lattice(E(T),≤)is isomorphic to the lattice of faces of a rational polytope in someℝn. Using these results, it is shown that ifSis any connected semigroup with lattice of regularJ-classesU(S), then all maximal chains inU(S)have the same length.