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Posets :: standardMonomialPoset

standardMonomialPoset -- generates the poset of divisibility in the monomial basis of an ideal

Synopsis

Description

The standard monomial poset of a MonomialIdeal is the poset of monomials in the quotient with partial ordering given by divisibility.
i1 : R = QQ[x,y,z];
i2 : standardMonomialPoset monomialIdeal(x^2, y^2, z^2, x*y*z)

o2 = Poset{cache => CacheTable{}                                                                                                          }
           GroundSet => {1, x, x*y, x*z, y, y*z, z}
           RelationMatrix => | 1 1 1 1 1 1 1 |
                             | 0 1 1 1 0 0 0 |
                             | 0 0 1 0 0 0 0 |
                             | 0 0 0 1 0 0 0 |
                             | 0 0 1 0 1 1 0 |
                             | 0 0 0 0 0 1 0 |
                             | 0 0 0 1 0 1 1 |
           Relations => {{1, x}, {1, x*y}, {1, x*z}, {1, y}, {1, y*z}, {1, z}, {x, x*y}, {x, x*z}, {y, x*y}, {y, y*z}, {z, x*z}, {z, y*z}}

o2 : Poset
If the integers minDeg and maxDeg are specified, then only the monomials with degrees between minDeg and maxDeg are used. As the standard monomial poset is ranked, this is the same as taking all the ranks between minDeg and maxDeg.
i3 : standardMonomialPoset(monomialIdeal(x^4, y^4, z^4, x*y*z), 3, 4)

o3 = Poset{cache => CacheTable{}                                                                                                                                                                                                                            }
                          3   3    3    2    2 2   2    2 2     2     3     2     3   3   3    2    2 2     2     3   3
           GroundSet => {x , x y, x z, x y, x y , x z, x z , x*y , x*y , x*z , x*z , y , y z, y z, y z , y*z , y*z , z }
           RelationMatrix => | 1 1 1 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 1 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                             | 0 0 1 0 0 1 1 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 1 0 0 1 1 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 1 0 0 1 1 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 1 0 0 1 1 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 1 1 1 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 1 1 1 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 1 0 0 0 0 0 1 1 |
                           3   3      3   3      2    3      2    2 2     2    3      2    2 2       2   2 2       2     3       2   2 2       2     3     3     3     3   3      2    3      2    2 2       2   2 2       2     3     3     3     3     3
           Relations => {{x , x y}, {x , x z}, {x y, x y}, {x y, x y }, {x z, x z}, {x z, x z }, {x*y , x y }, {x*y , x*y }, {x*z , x z }, {x*z , x*z }, {y , x*y }, {y , y z}, {y z, y z}, {y z, y z }, {y*z , y z }, {y*z , y*z }, {z , x*z }, {z , y*z }}

o3 : Poset

See also

Ways to use standardMonomialPoset :