High Quality Content by WIKIPEDIA articles! In combinatorial mathematics, the theory of Stanley–Reisner rings is a basic tool that supports the use of commutative algebra by defining a polynomial algebra for any abstract simplicial complex. Definition. An (augmented abstract) simplicial complex ? on a vertex set V? is a collection (possibly empty) of finite subsets ?, the simplices, of V? satisfying: mbox{(a)}text{ If } vin V_{Sigma}, text{ then } {v}in Sigma. mbox{(b)} text{ If } sigmain Sigma text{ and } tausubset sigma, text{ then } tauin Sigma. Define the simplicial join Sigma_1astSigma_2, of two simplicial complexes ?1 and ?2 with disjoint vertex sets, to be: Sigma_1astSigma_2:={sigma_1cupsigma_2 mid sigma_iinSigma_i (i=1,2) }. Three levels can be identified: a simplicial complex is a set of simplices, which are finite sets of vertices.

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