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.