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.
