Order Dimension, Strong Bruhat Order and Lattice Properties for Posets

We determine the order dimension of the strong Bruhat order on finite Coxeter groups of types A, B and H. The order dimension is determined using a generalization of a theorem of Dilworth: dim(P ) = width(Irr(P )), whenever P satisfies a simple order-theoretic condition called here the dissective property (or “clivage” in [16, 21]). The result for dissective posets follows from an upper bound and lower bound on the dimension of any finite poset. The dissective property is related, via MacNeille completion, to the distributive property of lattices. We show a similar connection between quotients of the strong Bruhat order with respect to parabolic subgroups and lattice quotients.