Editing Abstract simplicial complex (section)

{{Short description|Mathematical object}}
[[Image:Simplicial complex example.svg|thumb|200px|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 [[subset]]s, i.e., every subset of a set in the family is also in the family. It is a purely combinatorial description of the geometric notion of a [[simplicial complex]].<ref name=Lee>[[John M. Lee|Lee, John M.]], Introduction to Topological Manifolds, Springer 2011, {{ISBN|1-4419-7939-5}}, p153</ref> For example, in a 2-dimensional simplicial complex, the sets in the family are the triangles (sets of size 3), their edges (sets of size 2), and their vertices (sets of size 1).

In the context of [[matroid]]s and [[greedoid]]s, abstract simplicial complexes are also called '''[[independence system]]s'''.<ref>{{cite book|author = Korte, Bernhard|author-link = Bernhard Korte|author2=Lovász, László|author2-link=László Lovász|author3=Schrader, Rainer| year = 1991| title = Greedoids | publisher = Springer-Verlag | isbn = 3-540-18190-3 |page = 9}}</ref>

An abstract simplex can be studied algebraically by forming its [[Stanley–Reisner ring]]; this sets up a powerful relation between [[combinatorics]] and [[commutative algebra]].