Abstract simplicial complex

In mathematics, given a universal set S, and a family of sets K over S, K is an abstract simplicial complex if the following is true:

∀ X ⊆ S, if X ∈ K, then ∀ Y ⊂ X it follows that Y ∈ K.

The elements of K are called abstract simplices. Furthermore, for X ∈ K, define the dimension to be dim(X) = |X| − 1, and consequently define dim(K) = max{dim(X), X ∈ K}. One-dimensional simplicial complexes are called (simple) graphs. Finally, the K^(d) = {X ∈ K, dim(X) ≤ d} is called the d-skeleton of K. In particular, the skeleton is called the underlying graph.

See also

• simplicial complex,
• Kruskal-Katona theorem