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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.〔Lee, JM, Introduction to Topological Manifolds, Springer 2011, ISBN 1-4419-7939-5, p153〕 In the context of matroids and greedoids, abstract simplicial complexes are also called independence systems. ==Definitions== A family of non-empty finite subsets of a universal set ''S'' is an abstract simplicial complex if, for every set in , and every non-empty subset , also belongs to . The finite sets that belong to are called faces of the complex, and a face is said to belong to another face if , so the definition of an abstract simplicial complex can be restated as saying that every face of a face of a complex is itself a face of . The vertex set of is defined as , the union of all faces of . The elements of the vertex set are called the vertices of the complex. So for every vertex ''v'' of , the set is a face of the complex. The maximal faces of (i.e., faces that are not subsets of any other faces) are called facets of the complex. The dimension of a face in is defined as : faces consisting of a single element are zero-dimensional, faces consisting of two elements are one-dimensional, etc. The dimension of the complex 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. The complex is said to be finite if it has finitely many faces, or equivalently if its vertex set is finite. Also, is said to be pure if it is finite-dimensional (but not necessarily finite) and every facet has the same dimension. In other words, is pure if is finite and every face is contained in a facet of dimension . One-dimensional abstract simplicial complexes are mathematically equivalent to simple undirected graphs: the vertex set of the complex can be viewed as the vertex set of a graph, and the two-element facets of the complex correspond to undirected edges of a graph. In this view, one-element facets of a complex correspond to isolated vertices that do not have any incident edges. A subcomplex of is a simplicial complex ''L'' such that every face of ''L'' belongs to ; that is, and ''L'' is a simplicial complex. A subcomplex that consists of all of the subsets of a single face of is often called a simplex of . (However, some authors use the term "simplex" for a face or, rather ambiguously, for both a face and the subcomplex associated with a face, by analogy with the non-abstract (geometric) simplicial complex terminology. To avoid ambiguity, we do not use in this article the term "simplex" for a face in the context of abstract complexes.) The d-skeleton of is the subcomplex of consisting of all of the faces of that have dimension at most ''d''. In particular, the 1-skeleton is called the underlying graph of . The 0-skeleton of can be identified with its vertex set, although formally it is not quite the same thing (the vertex set is a single set of all of the vertices, while the 0-skeleton is a family of single-element sets). The link of a face in , often denoted or , is the subcomplex of defined by : Note that the link of the empty set is itself. Given two abstract simplicial complexes, and , a simplicial map is a function that maps the vertices of to the vertices of Γ and that has the property that for any face of , the image set is a face of . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「abstract simplicial complex」の詳細全文を読む スポンサード リンク
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