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Posets :: lcmLattice

lcmLattice -- generates the lattice of lcms in an ideal

Synopsis

Description

The LCM lattice of a MonomialIdeal is the set of all LCMs of subsets of the generators of the ideal with partial ordering given by divisbility. These are particularly useful in the study of resolutions of monomial ideals.
i1 : R = QQ[x,y];
i2 : lcmLattice monomialIdeal(x^2, x*y, y^2)

o2 = Poset{cache => CacheTable{}                                                                                                                                                                    }
                             2          2   2   2    2 2
           GroundSet => {1, y , x*y, x*y , x , x y, x y }
           RelationMatrix => | 1 1 1 1 1 1 1 |
                             | 0 1 0 1 0 0 1 |
                             | 0 0 1 1 0 1 1 |
                             | 0 0 0 1 0 0 1 |
                             | 0 0 0 0 1 1 1 |
                             | 0 0 0 0 0 1 1 |
                             | 0 0 0 0 0 0 1 |
                              2                    2        2        2         2 2     2     2     2   2 2            2          2           2 2       2   2 2     2   2      2   2 2     2    2 2
           Relations => {{1, y }, {1, x*y}, {1, x*y }, {1, x }, {1, x y}, {1, x y }, {y , x*y }, {y , x y }, {x*y, x*y }, {x*y, x y}, {x*y, x y }, {x*y , x y }, {x , x y}, {x , x y }, {x y, x y }}

o2 : Poset
If a non-monomial ideal is passed in, then the monomialIdeal of the ideal is used instead.
i3 : S = QQ[a,b,c,d];
i4 : lcmLattice ideal (b^2-a*d, a*d-b*c, c^2-b*d)

o4 = Poset{cache => CacheTable{}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     }
                             2          2   2   2    2 2            2                        2      2      2        2 2    2 2   2   2   2 2 2   2   2   2     2   2   2 2   2 2 2   2 2   2   2 2 2 2
           GroundSet => {1, c , b*c, b*c , b , b c, b c , a*c*d, a*c d, a*b*d, a*b*c*d, a*b*c d, a*b d, a*b c*d, a*b c d, a d , a c*d , a c d , a b*d , a b*c*d , a b*c d , a b d , a b c*d , a b c d }
           RelationMatrix => | 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 0 0 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 |
                             | 0 0 1 1 0 1 1 0 0 0 1 1 0 1 1 0 0 0 0 1 1 0 1 1 |
                             | 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 |
                             | 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 |
                             | 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 |
                             | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 |
                             | 0 0 0 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 |
                             | 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 |
                             | 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 |
                             | 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 1 1 0 1 1 |
                             | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 |
                             | 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 |
                             | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 |
                             | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 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 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 0 1 1 0 1 1 |
                             | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 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 |
                             | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 |
                             | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 |
                             | 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 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 |
                              2                    2        2        2         2 2                      2                                       2           2           2             2 2         2 2        2   2        2 2 2        2   2        2     2        2   2 2        2 2 2        2 2   2        2 2 2 2     2     2     2   2 2     2     2      2       2      2     2 2      2   2 2 2     2   2   2 2     2   2 2 2 2            2          2           2 2                              2             2               2 2           2     2          2   2 2          2 2   2          2 2 2 2       2   2 2       2       2        2     2 2        2   2   2 2       2   2 2 2 2     2   2      2   2 2     2     2      2     2        2     2 2      2   2 2 2     2   2 2   2     2   2 2 2 2     2    2 2     2      2        2      2 2      2    2 2   2     2    2 2 2 2     2 2     2 2      2 2   2 2 2 2              2                                   2               2                 2 2             2   2            2 2 2            2     2            2   2 2            2 2   2            2 2 2 2       2        2        2      2 2        2    2 2 2       2    2   2 2       2    2 2 2 2                                  2               2               2                 2 2             2   2            2     2            2   2 2            2 2 2            2 2   2            2 2 2 2                  2                 2                   2 2               2     2              2   2 2              2 2   2              2 2 2 2         2      2 2          2    2   2 2         2    2 2 2 2       2      2          2      2 2        2    2 2 2       2    2 2   2       2    2 2 2 2       2        2 2        2      2 2   2       2      2 2 2 2       2 2    2 2 2 2     2 2   2   2     2 2   2 2 2     2 2   2   2     2 2   2     2     2 2   2   2 2     2 2   2 2 2     2 2   2 2   2     2 2   2 2 2 2     2   2   2 2 2     2   2   2     2     2   2   2   2 2     2   2   2 2   2     2   2   2 2 2 2     2 2 2   2   2 2     2 2 2   2 2 2 2     2   2   2     2     2   2   2   2 2     2   2   2 2 2     2   2   2 2   2     2   2   2 2 2 2     2     2   2   2 2     2     2   2 2   2     2     2   2 2 2 2     2   2 2   2 2 2 2     2 2 2   2 2   2     2 2 2   2 2 2 2     2 2   2   2 2 2 2
           Relations => {{1, c }, {1, b*c}, {1, b*c }, {1, b }, {1, b c}, {1, b c }, {1, a*c*d}, {1, a*c d}, {1, a*b*d}, {1, a*b*c*d}, {1, a*b*c d}, {1, a*b d}, {1, a*b c*d}, {1, a*b c d}, {1, a d }, {1, a c*d }, {1, a c d }, {1, a b*d }, {1, a b*c*d }, {1, a b*c d }, {1, a b d }, {1, a b c*d }, {1, a b c d }, {c , b*c }, {c , b c }, {c , a*c d}, {c , a*b*c d}, {c , a*b c d}, {c , a c d }, {c , a b*c d }, {c , a b c d }, {b*c, b*c }, {b*c, b c}, {b*c, b c }, {b*c, a*b*c*d}, {b*c, a*b*c d}, {b*c, a*b c*d}, {b*c, a*b c d}, {b*c, a b*c*d }, {b*c, a b*c d }, {b*c, a b c*d }, {b*c, a b c d }, {b*c , b c }, {b*c , a*b*c d}, {b*c , a*b c d}, {b*c , a b*c d }, {b*c , a b c d }, {b , b c}, {b , b c }, {b , a*b d}, {b , a*b c*d}, {b , a*b c d}, {b , a b d }, {b , a b c*d }, {b , a b c d }, {b c, b c }, {b c, a*b c*d}, {b c, a*b c d}, {b c, a b c*d }, {b c, a b c d }, {b c , a*b c d}, {b c , a b c d }, {a*c*d, a*c d}, {a*c*d, a*b*c*d}, {a*c*d, a*b*c d}, {a*c*d, a*b c*d}, {a*c*d, a*b c d}, {a*c*d, a c*d }, {a*c*d, a c d }, {a*c*d, a b*c*d }, {a*c*d, a b*c d }, {a*c*d, a b c*d }, {a*c*d, a b c d }, {a*c d, a*b*c d}, {a*c d, a*b c d}, {a*c d, a c d }, {a*c d, a b*c d }, {a*c d, a b c d }, {a*b*d, a*b*c*d}, {a*b*d, a*b*c d}, {a*b*d, a*b d}, {a*b*d, a*b c*d}, {a*b*d, a*b c d}, {a*b*d, a b*d }, {a*b*d, a b*c*d }, {a*b*d, a b*c d }, {a*b*d, a b d }, {a*b*d, a b c*d }, {a*b*d, a b c d }, {a*b*c*d, a*b*c d}, {a*b*c*d, a*b c*d}, {a*b*c*d, a*b c d}, {a*b*c*d, a b*c*d }, {a*b*c*d, a b*c d }, {a*b*c*d, a b c*d }, {a*b*c*d, a b c d }, {a*b*c d, a*b c d}, {a*b*c d, a b*c d }, {a*b*c d, a b c d }, {a*b d, a*b c*d}, {a*b d, a*b c d}, {a*b d, a b d }, {a*b d, a b c*d }, {a*b d, a b c d }, {a*b c*d, a*b c d}, {a*b c*d, a b c*d }, {a*b c*d, a b c d }, {a*b c d, a b c d }, {a d , a c*d }, {a d , a c d }, {a d , a b*d }, {a d , a b*c*d }, {a d , a b*c d }, {a d , a b d }, {a d , a b c*d }, {a d , a b c d }, {a c*d , a c d }, {a c*d , a b*c*d }, {a c*d , a b*c d }, {a c*d , a b c*d }, {a c*d , a b c d }, {a c d , a b*c d }, {a c d , a b c d }, {a b*d , a b*c*d }, {a b*d , a b*c d }, {a b*d , a b d }, {a b*d , a b c*d }, {a b*d , a b c d }, {a b*c*d , a b*c d }, {a b*c*d , a b c*d }, {a b*c*d , a b c d }, {a b*c d , a b c d }, {a b d , a b c*d }, {a b d , a b c d }, {a b c*d , a b c d }}

o4 : Poset

See also

Ways to use lcmLattice :