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# star(SimplicialComplex,RingElement) -- make the star of a face

## Synopsis

• Function: star
• Usage:
star(Delta, F)
• Inputs:
• Delta, ,
• F, , a monomial representing a subset of the vertices in $\Delta$
• Outputs:
• , that consists of all faces in $\Delta$ whose union with $F$ is also a face in $\Delta$

## Description

Given a subset $F$ of the vertices in an abstract simplicial complex $\Delta$, the star of $F$ is the set of faces $G$ in $\Delta$ such that the union of $G$ and $F$ is also a face in $\Delta$. This set forms a subcomplex of $\Delta$. When the subset $F$ is not face in $\Delta$, the star of $F$ is a void complex (having no facets).

The star of a subset $F$ may be the entire complex, a proper subcomplex, or the void complex.

 i1 : S = ZZ[a..e]; i2 : Δ = simplicialComplex {a*b*c, c*d*e} o2 = simplicialComplex | cde abc | o2 : SimplicialComplex i3 : star (Δ, c) o3 = simplicialComplex | cde abc | o3 : SimplicialComplex i4 : assert (star (Δ, c) === Δ) i5 : star (Δ, a*b) o5 = simplicialComplex | abc | o5 : SimplicialComplex i6 : assert (star (Δ, a*b) === simplicialComplex {a*b*c}) i7 : star (Δ, a*d) o7 = simplicialComplex 0 o7 : SimplicialComplex i8 : assert (star (Δ, a*d) === simplicialComplex monomialIdeal 1_S)