Abstract simplicial complex

In mathematics, an abstract simplicial complex is a purely combinatorial description of the geometric notion of a simplicial complex, consisting of a family of non-empty finite sets closed under the operation of taking non-empty subsets. In the context of matroids and greedoids, abstract simplicial complexes are also called independence systems.

A family $Δ$ of non-empty finite subsets of a set S is an abstract simplicial complex if, for every set $X$ in $Δ$ , and every non-empty subset $Y \subset X$ , $Y$ also belongs to $Δ$ .

The finite sets that belong to $Δ$ are called faces of the complex, and a face $Y$ is said to belong to another face $X$ if $Y \subset X$ , so the definition of an abstract simplicial complex can be restated as saying that every face of a face of a complex $Δ$ is itself a face of $Δ$ . The vertex set of $Δ$ is defined as $V (Δ) = \cupΔ$ , the union of all faces of $Δ$ . The elements of the vertex set are called the vertices of the complex. So for every vertex v of $Δ$ , the set {v} is a face of the complex. The maximal faces of $Δ$ (i.e., faces that are not subsets of any other faces) are called facets of the complex. The dimension of a face $X$ in $Δ$ is defined as $dim(X) = | X | - 1$ : faces consisting of a single element are zero-dimensional, faces consisting of two elements are one-dimensional, etc. The dimension of the complex $dim(Δ)$ is defined as the largest dimension of any of its faces, or infinity if there is no finite bound on the dimension of the faces.

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