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}.

Define a simplicial complex K with vertices the maximal elements of P and with simplices those subsets with a common lower bound.

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.

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].

i)] of

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

Recall that a simplicial map f from

a simplicial complex [GAMMA] to a poset P sends vertices of [GAMMA] to elements of P and faces of the simplicial complex to chains of P.

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

a simplicial complex.

Instead of being defined on the topological space of a geometric realization of

a simplicial complex, the discrete homotopy groups are defined in terms of the combinatorial connectivity of the complex.