For a finite geometric left regular band B, we will use the following special case of Rota's cross-cut theorem [26, 6] to provide

a simplicial complex homotopy equivalent to the order complex [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of B\{1}.

Figure 1 displays

a simplicial complex, where some edges are "collapsed".

A typical discrete representation of such a subspace is as

a simplicial complex embedded in [R.sup.3], and of a function is as a simplicial map between complexes.

Note that a polytope P is simplicial if and only if the boundary of P is

a simplicial complex. The elements of

a simplicial complex are also called faces and the dimension of

a simplicial complex is the maximal dimension of a face.

In the following, let [DELTA] be

a simplicial complex whose Stanley-Reisner ideal appears as an initial ideal of [I.sub.k,n].

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]).

For

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

A simplicial complex [DELTA] on a finite ground set E is a collection of subsets of E such that if S [member of] [DELTA] and T [subset or equal to] S, then T [member of] [DELTA].

The i-dimensional skeleton [[DELTA].sub.(i)] of

a simplicial complex [DELTA] is the subcomplex consisting of all cells of dimension [less than or equal to] i.

Since the permutahedron is a simple polytope the complex [P.sub.n] is

a simplicial complex homeomorphic to an (n - 2)-dimensional sphere.

Section 4 recalls results and establishes terminology on Kalai's higher dimensional spanning trees in

a simplicial complex. Section 5 discusses further properties of the simplicial complex [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] whose subcomplexes appear in Theorem 1 and 2.