# Complex -- The class of all simplicial complexes.

## Description

The class of all simplicial complexes on the variables of a PolynomialRing. A complex is represented by its maximal faces.

A complex can be constructed from a reduced monomial Ideal by the method idealToComplex

 `i1 : K=QQ;` `i2 : R=K[x_0..x_4];` ```i3 : I=ideal(x_0*x_1,x_1*x_2,x_2*x_3,x_3*x_4,x_4*x_0); o3 : Ideal of R``` ```i4 : D =idealToComplex I o4 = {x x , x x , x x , x x , x x } 2 4 0 3 0 2 1 3 1 4 o4 : complex with 5 facets on the vertices x x x x x 0 1 2 3 4``` ```i5 : simplexRing D o5 = R o5 : PolynomialRing``` ```i6 : dimension D o6 = 1``` ```i7 : L=facets D o7 = {x x , x x , x x , x x , x x } 2 4 0 3 0 2 1 3 1 4 o7 : List``` ```i8 : D==complex L o8 = true``` ```i9 : complexToIdeal D o9 = ideal (x x , x x , x x , x x , x x ) 3 4 0 4 2 3 1 2 0 1 o9 : Ideal of R```

• idealToComplex -- Compute the Stanley-Reisner complex.
• complexToIdeal -- Compute the Stanley-Reisner ideal.
• facets -- The facets of a simplicial complex.
• faces -- Returns the faces of a complex
• Face -- The class of faces of simplicial complexes.

## Methods that use an object of class Complex :

• Complex == Complex -- Compare two complexes.
• complexToIdeal(Complex), see complexToIdeal -- Compute the Stanley-Reisner ideal.
• dimension(Complex), see dimension -- The dimension of a simplicial complex or a face of a simplicial complex.
• faces(Complex), see faces -- Returns the faces of a complex
• faces(Complex,ZZ), see faces -- Returns the faces of a complex
• facets(Complex), see facets -- The facets of a simplicial complex.
• fvector(Complex), see fvector -- Returns the F-vector of a complex
• simplexRing(Complex), see simplexRing -- The underlying polynomial ring of a simplicial complex, face, fan or cone.
• substituteComplex(Complex,PolynomialRing), see substituteComplex -- Substitute a complex to a different ring.

## For the programmer

The object Complex is a type, with ancestor classes List < VisibleList < BasicList < Thing.