Since the theory of hypergraphs is still too under-developed, we resort to geometry and topology, which view a graph as a one-dimensional

simplicial complex. I want to develop a combinatorial/geometric/probabilistic theory of higher-dimensional

simplicial complexes.

An abstract

simplicial complex [DELTA] is a collection of sets such that B [member of] [DELTA] for any subset B with B [subset or equal to] A [member of] A.

A

simplicial complex in 4-d corresponds to a set of simplices of p = 0, ..., 4 degrees comprising an irregular (in general) tessellation of spacetime, i.e., where the boundaries of each p-simplex comprise a set of lower dimensional, (p - 1)-simplices, for p = 1, ..., 4 (that is, with no overlaps or gaps).

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}.

The topological model is a

simplicial complex, which is a topological space formed by gluing together vertices, edges, filled triangular faces, solid tetrahedra, and higher dimensional analogues of these convex polytopes according to a few rules about how the gluing is allowed to be done [25].

[5] does not mention the absence of local cut vertices or that the grid must be conforming, however the definition of a grid in that paper is that it be a "

simplicial complex coming from the simplicial decomposition of a connected 2D manifold" which implies these conditions.

A digital subcomplex A of a digital

simplicial complex X with [kappa]-adjacency is a digital

simplicial complex [15] contained in X with Vert(A) [subset] Vert(X).

Define a

simplicial complex [DELTA]([PHI]) as the set of all subsets A [subset or equal to] [[PHI].sub.[greater than or equal to]-1] such that all almost positive roots in A are pairwise compatible.

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.

Additional Key Words and Phrases: Algebraic topology, asynchronous distributed computation, decision tasks, distributed computing, homology,

simplicial complex, wait-free algorithms

Then the lower envelope of conv([??]) is combinatorially equivalent (as a

simplicial complex) to the placing triangulation of A.

Figure 1 displays a

simplicial complex, where some edges are "collapsed".