Shellability in reductive monoids

The purpose of this paper is to extend to monoids the work of Björner, Wachs and Proctor on the shellability of the Bruhat-Chevalley order on Weyl groups. Let be a reductive monoid with unit group , Borel subgroup and Weyl group . We study the partially ordered set of -orbits (with respect to Zariski closure inclusion) within a -orbit of . This is the same as studying a -orbit in the Renner monoid . Such an orbit is the retract of a `universal orbit', which is shown to be lexicograhically shellable in the sense of Björner and Wachs.