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  2. Abstract simplicial complex - Wikipedia

    en.wikipedia.org/wiki/Abstract_simplicial_complex

    Geometric realization of a 3-dimensional abstract simplicial complex. 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.

  3. 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 , ≤).

  4. Alexander duality - Wikipedia

    en.wikipedia.org/wiki/Alexander_duality

    Let be an abstract simplicial complex on a vertex set of size . The Alexander dual X ∗ {\displaystyle X^{*}} of X {\displaystyle X} is defined as the simplicial complex on V {\displaystyle V} whose faces are complements of non-faces of X {\displaystyle X} .

  5. h-vector - Wikipedia

    en.wikipedia.org/wiki/H-vector

    Let Δ be an abstract simplicial complex of dimension d − 1 with f i i-dimensional faces and f −1 = 1. These numbers are arranged into the f-vector of Δ, = (,, …,).An important special case occurs when Δ is the boundary of a d-dimensional convex polytope.

  6. Category:Simplicial sets - Wikipedia

    en.wikipedia.org/wiki/Category:Simplicial_sets

    Download QR code; Print/export Download as PDF; Printable version; In other projects Wikidata item; Appearance. ... Abstract simplicial complex; C. Clique complex; D ...

  7. 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.

  8. Triangulation (topology) - Wikipedia

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

    A natural question is if vice versa, any abstract simplicial complex corresponds to a geometric complex. In general, the geometric construction as mentioned here is not flexible enough: consider for instance an abstract simplicial complex of infinite dimension.

  9. Hypergraph - Wikipedia

    en.wikipedia.org/wiki/Hypergraph

    A downward-closed hypergraph is usually called an abstract simplicial complex. It is generally not reduced, unless all hyperedges have cardinality 1. An abstract simplicial complex with the augmentation property is called a matroid. Laminar: for any two hyperedges, either they are disjoint, or one is included in the other.