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# isPure(SimplicialComplex) -- whether the facets are equidimensional

## Synopsis

• Function: isPure
• Usage:
isPure Delta
• Inputs:
• Delta, ,
• Outputs:
• , 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