# monomialIdeal(SimplicialComplex) -- the monomial ideal of minimal nonfaces (the Stanley-Reisner ideal)

## Synopsis

• Function: monomialIdeal
• Usage:
monomialIdeal D
• Inputs:
• Outputs:
• , which is generated by monomials representing the minimal nonfaces of D

## Description

In Macaulay2, every simplicial complex is equipped with a polynomial ring, and the Stanley-Reisner ideal is contained in this ring.

The 3-dimensional sphere has a unique minimal nonface which corresponds to the interior.

 i1 : R = ZZ[a..e]; i2 : sphere = simplicialComplex {b*c*d*e,a*c*d*e,a*b*d*e,a*b*c*e,a*b*c*d} o2 = | bcde acde abde abce abcd | o2 : SimplicialComplex i3 : monomialIdeal sphere o3 = monomialIdeal(a*b*c*d*e) o3 : MonomialIdeal of R
The simplicial complex from example 1.8 in Miller-Sturmfels, Combinatorial Commutative Algebra, consists of a triangle (on vertices a,b,c), two edges connecting c to d and b to d, and an isolated vertex e.
 i4 : D = simplicialComplex {e, c*d, b*d, a*b*c} o4 = | e cd bd abc | o4 : SimplicialComplex i5 : monomialIdeal D o5 = monomialIdeal (a*d, b*c*d, a*e, b*e, c*e, d*e) o5 : MonomialIdeal of R
There are six minimal nonfaces of D.

This routine is identical to ideal(SimplicialComplex), except for the type of the output.

Note that no computatation is performed by this routine; all the computation was done while constructing the simplicial complex.