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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.
A complex is a structure in the unconscious that is objectified as an underlying theme—like a power or a status—by grouping clusters of emotions, memories, perceptions and wishes in response to a threat to the stability of the self.
Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex .
Let K be an abstract simplicial complex (ASC). The face poset of K is a poset made of all nonempty simplices of K , ordered by inclusion (which is a partial order). For example, the face-poset of the closure of {A,B,C} is the poset with the following chains:
Cognitive complexity is a psychological characteristic or psychological variable that indicates how complex or simple is the frame and perceptual skill of a person.. A person who is measured high on cognitive complexity tends to perceive nuances and subtle differences while a person with a lower measure, indicating a less complex cognitive structure for the task or activity, does not.
Simplicial homology is defined by a simple recipe for any abstract simplicial complex. It is a remarkable fact that simplicial homology only depends on the associated topological space. [2]: sec.8.6 As a result, it gives a computable way to distinguish one space from another.
Every flag complex is a clique complex: given a flag complex, define a graph G on the set of all vertices, where two vertices u,v are adjacent in G iff {u,v} is in the complex (this graph is called the 1-skeleton of the complex). By definition of a flag complex, every set of vertices that are pairwise-connected, is in the complex.
The dimension of an abstract simplicial complex is defined as () = {():}. [ 1 ] Abstract simplicial complexes can be realized as geometrical objects by associating each abstract simplex with a geometric simplex, defined below.