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Concept
Abstract simplicial complex
Summary
In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. It is a purely combinatorial description of the geometric notion of a simplicial complex. For example, in a 2-dimensional simplicial complex, the sets in the family are the triangles (sets of size 3), their edges (sets of size 2), and their vertices (sets of size 1).
In the context of matroids and greedoids, abstract simplicial complexes are also called independence systems.
An abstract simplex can be studied algebraically by forming its Stanley–Reisner ring; this sets up a powerful relation between combinatorics and commutative algebra.
A collection Δ of non-empty finite subsets of a set S is called a set-family.
A set-family Δ is called an abstract simplicial complex if, for every set X in Δ, and every non-empty subset Y ⊆ X, the set 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 ⊆ 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) = ∪Δ, the union of all faces of Δ. The elements of the vertex set are called the vertices of the complex. For every vertex v of Δ, the set {v} is a face of the complex, and every face of the complex is a finite subset of the vertex set.
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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