isSimplicialMA -- Test whether a monomial algebra is simplicial.

Synopsis

Usage:

isSimplicialMA R

isSimplicialMA B

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

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 a method function.