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SimplicialComplexes :: ideal(SimplicialComplex)

ideal(SimplicialComplex) -- the ideal of minimal nonfaces (the Stanley-Reisner ideal)



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 : ideal sphere

o3 = ideal(a*b*c*d*e)

o3 : Ideal 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 : ideal D

o5 = ideal (a*d, b*c*d, a*e, b*e, c*e, d*e)

o5 : Ideal of R
There are six minimal nonfaces of D.

This routine is identical to monomialIdeal(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.

See also