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# isSimplicialMA -- Test whether a monomial algebra is simplicial.

## Synopsis

• Usage:
isSimplicialMA R
isSimplicialMA B
isSimplicialMA M
• Inputs:
• R, , with B = degrees R and K = coefficientRing R, or
• B, a list, with the generators of an affine semigroup in \mathbb{N}^d.
• M, ,
• Outputs:

## Description

Test whether the monomial algebra K[B] is simplicial, that is, the cone C(B) is spanned by linearly independent vectors.

Note that this condition does not depend on K.

 i1 : B={{1,0,0},{0,2,0},{0,0,2},{1,0,1},{0,1,1}} o1 = {{1, 0, 0}, {0, 2, 0}, {0, 0, 2}, {1, 0, 1}, {0, 1, 1}} o1 : List i2 : R=QQ[x_0..x_4,Degrees=>B] o2 = R o2 : PolynomialRing i3 : isSimplicialMA R o3 = true i4 : isSimplicialMA B o4 = true

 i5 : B={{1,0,1},{0,1,1},{1,1,1},{0,0,1}} o5 = {{1, 0, 1}, {0, 1, 1}, {1, 1, 1}, {0, 0, 1}} o5 : List i6 : R=QQ[x_0..x_3,Degrees=>B] o6 = R o6 : PolynomialRing i7 : isSimplicialMA R o7 = false i8 : isSimplicialMA B o8 = false

 i9 : B={{1,0,1},{0,1,1},{1,1,1},{0,0,1}} o9 = {{1, 0, 1}, {0, 1, 1}, {1, 1, 1}, {0, 0, 1}} o9 : List i10 : M=monomialAlgebra B ZZ o10 = ---[x ..x ] 101 0 3 o10 : MonomialAlgebra generated by {{1, 0, 1}, {0, 1, 1}, {1, 1, 1}, {0, 0, 1}} i11 : isSimplicialMA M o11 = false

## Ways to use isSimplicialMA :

• isSimplicialMA(List)
• isSimplicialMA(MonomialAlgebra)
• isSimplicialMA(PolynomialRing)

## For the programmer

The object isSimplicialMA is .