Let [DELTA](B[X, Y)) be the order complex of the poset B[X, Y]\{y}: it is the

simplicial complex whose vertex set is B[X, Y]\{y} and whose simplices are the finite chains in the poset.

Viewed as a CW-complex, M then has the same homotopy type of a

simplicial complex which affords further considerations particularly when reducing matters to a skeletal-like, graph-theoretic analysis.

Define a

simplicial complex [DELTA]([PHI]) as the set of all subsets A [subset or equal to] [[PHI].

A (geometric)

simplicial complex [DELTA] is a finite family of simplices such that (i) if F is in [DELTA] and G is a face of F, then G is also in [DELTA], and (ii) for any two elements F and H of [DELTA], F [intersection] H is a face of both F and H.

More generally, the face lattice of its polar is isomorphic to the

simplicial complex of crossing-free subsets of internal diagonals of the (d + 3)-gon.

Recall here that a flag

simplicial complex is the complex of cliques of a graph.

The primary difference between a simplicial poset and a

simplicial complex is that any pair of faces in a

simplicial complex intersect along a single (possibly empty) face of their boundaries; whereas a pair of faces in a simplicial poset can intersect along any sub complex of their boundaries.

If E is a graph, then M([SIGMA]) is its usual graphic matroid, while if E is a

simplicial complex then M (E) is its simplicial matroid (see Cordovil and Lindstrom [1987]).

The collection of c-clusters forms a

simplicial complex on the set [[PHI].

In Section 6 the cd-index of the nth semisuspension of a non-pure shellable

simplicial complex is determined.

For a

simplicial complex [DELTA] and its face F [member of] [DELTA], the link of F in [DELTA] is the

simplicial complexMiller [KM04] define the subword complex SC (Q, [rho]) to be the

simplicial complex of those subwords of Q whose complements contain a reduced expression for [rho] as a subword.