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Making an abstract simplicial complex -- information about the basic constructors

An abstract simplicial complex on a finite set is a family of subsets that is closed under taking subsets. The elements in the abstract simplicial complex are called its faces. The faces having cardinality 1 are its vertices and the maximal faces (order by inclusion) are its facets. Following the combinatorial conventions, every nonempty abstract simplicial complex has the empty set as a face.

In this package, a simplicial complex is represented by its Stanley–Reisner ideal. The vertices are identified with a subset of the variables in a polynomial ring and each face is identified with the product of the corresponding variables. A nonface is any subset of the variables that does not belong to the simplicial complex and each nonface is again identified with product of variables. The Stanley-Reisner ideal of a simplicial complex is generated by monomials corresponding to its nonfaces.

Basic constructors for abstract simplicial complexes

Classic examples of abstract simplicial complexes

Other operations producing abstract simplicial complexes

See also