getPoset -- calculates the monomial poset of a ring

Usage:

getPoset(R, I)

getPoset I

getPoset S

getPoset R
Inputs:
- R, a polynomial ring,
- I, an ideal, a homogeneous ideal of a polynomial ring
- S, a quotient ring, a quotient of a polynomial ring R and a homogeneous ideal I of R
Optional inputs:
- MaxDegree => a number, default value 10, the maximum degree of a monomial to calculate
Outputs:
- an instance of the type Poset,

Description

Given a polynomial ring R and a homogeneous ideal of R (an ideal generated by homogeneous elements of R) called I, the monomial poset of R/I is the collection of monomials of R/I equipped with the divisibility partial order.

i1 : R = QQ[x,y]

o1 = R

o1 : PolynomialRing

i2 : I = ideal(x*y - y^2, x^4, x^3*y, x^2*y^2)

                   2   4   3    2 2
o2 = ideal (x*y - y , x , x y, x y )

o2 : Ideal of R

i3 : getPoset(R, I)

o3 = Relation Matrix: | 1 1 1 1 1 1 1 |
                      | 0 1 1 1 0 1 1 |
                      | 0 0 1 1 0 0 1 |
                      | 0 0 0 1 0 0 0 |
                      | 0 0 0 0 1 1 1 |
                      | 0 0 0 0 0 1 1 |
                      | 0 0 0 0 0 0 1 |

o3 : Poset

For posets with infinitely many monomials, the optional argument MaxDegree can be used to calculate monomials up to a certain degree to use in the poset. If this is left blank, the default value used will be 10.

i4 : J = ideal(x^2 - y^2)

            2    2
o4 = ideal(x  - y )

o4 : Ideal of R

i5 : getPoset(R, J, MaxDegree => 15)
Ideal is not finite over the base (quotient ring has infinitely many monomials)
Monomials up to degree 15 given

o5 = Relation Matrix: | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
                      | 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
                      | 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
                      | 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
                      | 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
                      | 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
                      | 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
                      | 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
                      | 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
                      | 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |

o5 : Poset

Caveat

Like the function getMons, the optional argument MaxDegree is only meant to be used for quotient rings where there are infinitely many monomials. For quotient rings with finitely many monomials, MaxDegree will be ignored regardless of it's value.

Ways to use getPoset:

getPoset(Ideal)
getPoset(PolynomialRing)
getPoset(PolynomialRing,Ideal)
getPoset(QuotientRing)

For the programmer

The object getPoset is a method function with options.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/MacaulayPosets.m2:1624:0.

getPoset -- calculates the monomial poset of a ring

Description

Caveat

See also

Ways to use getPoset:

For the programmer