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  2. Abstract simplicial complex - Wikipedia

    en.wikipedia.org/wiki/Abstract_simplicial_complex

    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 ...

  3. Triangulation (topology) - Wikipedia

    en.wikipedia.org/wiki/Triangulation_(topology)

    An abstract simplicial complex above a set is a system () of non-empty subsets such that: {} for each ;if and , then .; The elements of are called simplices, the elements of are called vertices.

  4. Poset topology - Wikipedia

    en.wikipedia.org/wiki/Poset_topology

    The order complex associated to a poset (S, ≤) has the set S as vertices, and the finite chains of (S, ≤) as faces. The poset topology associated to a poset ( S , ≤) is then the Alexandrov topology on the order complex associated to ( S , ≤).

  5. Field with one element - Wikipedia

    en.wikipedia.org/wiki/Field_with_one_element

    In 1957, Jacques Tits introduced the theory of buildings, which relate algebraic groups to abstract simplicial complexes.One of the assumptions is a non-triviality condition: If the building is an n‑dimensional abstract simplicial complex, and if k < n, then every k‑simplex of the building must be contained in at least three n‑simplices.

  6. Stanley–Reisner ring - Wikipedia

    en.wikipedia.org/wiki/Stanley–Reisner_ring

    Given an abstract simplicial complex Δ on the vertex set {x 1,...,x n} and a field k, the corresponding Stanley–Reisner ring, or face ring, denoted k[Δ], is obtained from the polynomial ring k[x 1,...,x n] by quotienting out the ideal I Δ generated by the square-free monomials corresponding to the non-faces of Δ:

  7. Kruskal–Katona theorem - Wikipedia

    en.wikipedia.org/wiki/Kruskal–Katona_theorem

    In algebraic combinatorics, the Kruskal–Katona theorem gives a complete characterization of the f-vectors of abstract simplicial complexes.It includes as a special case the Erdős–Ko–Rado theorem and can be restated in terms of uniform hypergraphs.

  8. Simplicial complex - Wikipedia

    en.wikipedia.org/wiki/Simplicial_complex

    A simplicial 3-complex. In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their n-dimensional counterparts (see illustration). ). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy th

  9. Simplicial complex recognition problem - Wikipedia

    en.wikipedia.org/wiki/Simplicial_complex...

    An abstract simplicial complex (ASC) is family of sets that is closed under taking subsets (the subset of a set in the family is also a set in the family). Every abstract simplicial complex has a unique geometric realization in a Euclidean space as a geometric simplicial complex (GSC), where each set with k elements in the ASC is mapped to a (k-1)-dimensional simplex in the GSC.