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- The number of abstract simplicial complexes on n elements is one less than the nth Dedekind number.
- See also the definition of an abstract simplicial complex, which loosely speaking is a simplicial complex without an associated geometry.
- The result of barycentric subdivision, when viewed as an abstract simplicial complex, is an example of a flag complex.
- An undirected graph can be viewed as an abstract simplicial complex C with a single-element set per vertex and a two-element set per edge.
- In the context of matroids and greedoids, abstract simplicial complexes are also called independence systems.
- Obviously, if J belongs to N, then any of its subsets is also in N. Therefore N is an abstract simplicial complex.
- The associated combinatorial structure is called an abstract simplicial complex, in which context the word "simplex" simply means any finite set of vertices.
- Simplicial sets should not be confused with abstract simplicial complexes, which generalize simple undirected graphs rather than directed multigraphs.
- The Dedekind numbers also count the number of abstract simplicial complexes on n elements, families of sets with the property that any subset of a set in the family also belongs to the family.
- The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex.
- In mathematics, the nerve of an open covering is a construction in topology, of an abstract simplicial complex from an open covering of a topological space X.
- An abstract simplicial complex Δ is called Cohen-Macaulay over k if its face ring is a Cohen-Macaulay ring.
- Special kinds of hypergraphs include, besides k-uniform ones, clutters, where no edge appears as a subset of another edge; and abstract simplicial complexes, which contain all subsets of every edge.
- By definition, such a space is homeomorphic to a simplicial complex (more precisely, the geometric realization of an abstract simplicial complex).
- The definition of h-vector applies to arbitrary abstract simplicial complexes.
- One of the assumptions is a non-triviality condition: If the building is an n-dimensional abstract simplicial complex, and if, then every k-simplex of the building must be contained in at least three n-simplices.
- In 1957, Jacques Tits introduced the theory of buildings, which relate algebraic groups to abstract simplicial complexes.
- 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 an abstract simplicial complex, a set S of vertices that is not itself part of the complex, but such that each pair of vertices in S belongs to some simplex in the complex, is called an empty simplex.
- Equivalently, it counts the number of antichains of subsets of an n-element set, the number of elements in a free distributive lattice with n generators, or the number of abstract simplicial complexes with n elements.