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...

counterpart to a simplicial complex is an abstract simplicial complex. To distinguish a simplicial complex from an abstract simplicial complex, the former...

the topological properties of simplicial complexes and their generalizations, cell-complexes. An abstract simplicial complex above a set V {\displaystyle...

A} and B {\displaystyle B} are any abstract simplicial complexes, then their join is an abstract simplicial complex defined as follows:: 74, Def.4.2.1 ...

mathematics, a simplicial set is a sequence of sets with internal order structure (abstract simplices) and maps between them. Simplicial sets are higher-dimensional...

information about the local structure of the complex at the vertex. Given an abstract simplicial complex X and v {\textstyle v} a vertex in V ( X ) {\textstyle...

: sec.5.3.2  Simplicial homology is defined by a simple recipe for any abstract simplicial complex. It is a remarkable fact that simplicial homology only...

simplices to form a simplicial complex. The geometric simplex and simplicial complex should not be confused with the abstract simplicial complex, in which a simplex...

subgraphs) of an undirected graph. The clique complex X(G) of an undirected graph G is an abstract simplicial complex (that is, a family of finite sets closed...

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...

making N ( C ) {\displaystyle N(C)} an abstract simplicial complex. Hence N(C) is often called the nerve complex of C {\displaystyle C} . Let X be the...

the sets as its vertices. An abstract simplicial complex is a combinatorial abstraction of the notion of a simplicial complex, a shape formed by unions of...

(S, ≤), ordered by inclusion. Let V be a set of vertices. An abstract simplicial complex Δ is a set of finite sets of vertices, known as faces σ ⊆ V {\displaystyle...

algebraic topology and topological data analysis, the Čech complex is an abstract simplicial complex constructed from a point cloud in any metric space which...

come from an abstract simplicial complex, because all three triangles share the same three vertices, while abstract simplicial complexes require each...

Steinitz is related to the notion of an abstract simplicial complex and it differs from a simplicial complex by the property that its elements are not...

refinement) of a simplicial complex is another simplicial complex in which, intuitively, one or more simplices of the original complex have been partitioned...

illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory...

another fixed simplicial complex. The problem is undecidable for complexes of dimension 5 or more.: 9–11  An abstract simplicial complex (ASC) is family...

Simplex Simplicial complex Polytope Triangulation Barycentric subdivision Simplicial approximation theorem Abstract simplicial complex Simplicial set Simplicial...

independence complex of an undirected graph G, denoted by I(G), is an abstract simplicial complex (that is, a family of finite sets closed under the operation...

illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory...

topological space from distances in a set of points. It is an abstract simplicial complex that can be defined from any metric space M and distance δ by...

Melvin Hochster, and Gerald Reisner in the early 1970s. Given an abstract simplicial complex Δ on the vertex set {x1,...,xn} and a field k, the corresponding...

A chessboard complex is a particular kind of abstract simplicial complex, which has various applications in topological graph theory and algebraic topology...

A simplicial map is defined in slightly different ways in different contexts. Let K and L be two abstract simplicial complexes (ASC). A simplicial map...

operation on simplicial complexes. In algebraic topology it is sometimes useful to replace the original spaces with simplicial complexes via triangulations:...

combinatorial structure given by a complex of groups. A complex of groups (Y,f,g) on an abstract simplicial complex Y is given by a finite group Γσ for...

non-negatively graded chain complexes over a fixed base ring. A simplicial set is an abstract generalization of a simplicial complex and can play a role of...

definition of h-vector applies to arbitrary abstract simplicial complexes. The g-conjecture stated that for simplicial spheres, all possible h-vectors occur...