isPure(SimplicialComplex) -- whether the facets are equidimensional

Synopsis

Function: isPure
Usage:

isPure Delta
Inputs:
- Delta, an abstract simplicial complex,
Outputs:
- a Boolean value, which is true if the facets of $\Delta$ are of the same dimension, and false otherwise.

Description

An abstract simplicial complex is pure of dimension $d$ if every facet has the same dimension.

Many classic examples of abstract simplicial complexes are pure.

i1 : S = ZZ[x_1..x_18];

i2 : isPure simplexComplex(5, S)

o2 = true

i3 : isPure bartnetteSphereComplex S

o3 = true

i4 : isPure bjornerComplex S

o4 = true

i5 : isPure dunceHatComplex S

o5 = true

i6 : isPure poincareSphereComplex S

o6 = true

The abstract simplicial complex from Example 1.8 of 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 (namely $e$). It has six minimal nonfaces. Moreover, its 1-skeleton and 2-skeleton are not pure.

i7 : R = ZZ/101[a..f];

i8 : Γ = simplicialComplex {e, c*d, b*d, a*b*c}

o8 = simplicialComplex | e cd bd abc |

o8 : SimplicialComplex

i9 : isPure Γ

o9 = false

i10 : isPure skeleton (1, Γ)

o10 = false

i11 : isPure skeleton (2, Γ)

o11 = false

There are two "trivial" simplicial complexes: the irrelevant complex has the empty set as a facet whereas the void complex has no faces. Both are pure.

i12 : irrelevant = simplicialComplex monomialIdeal gens S

o12 = simplicialComplex | 1 |

o12 : SimplicialComplex

i13 : isPure irrelevant

o13 = true

i14 : void = simplicialComplex monomialIdeal 1_S

o14 = simplicialComplex 0

o14 : SimplicialComplex

i15 : isPure void

o15 = true

Ways to use this method: