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Posets :: coveringRelations

coveringRelations -- returns a list of all relations (a < b) with no intermediates

Synopsis

Description

This command computes the list of all covering relations of a poset P. A relation (a,b) is said to be a covering relation if a < b and there is no element c with a < c < b. The result of this method is cached.
i1 : S = QQ[a,b,c,d];
i2 : P = divisorPoset(a*b^2*d);
i3 : P.GroundSet

                     2   2                          2     2
o3 = {1, d, b, b*d, b , b d, a, a*d, a*b, a*b*d, a*b , a*b d}

o3 : List
i4 : P.Relations

                                             2        2                     
o4 = {(1, 1), (1, d), (1, b), (1, b*d), (1, b ), (1, b d), (1, a), (1, a*d),
     ------------------------------------------------------------------------
                                  2          2                           2   
     (1, a*b), (1, a*b*d), (1, a*b ), (1, a*b d), (d, d), (d, b*d), (d, b d),
     ------------------------------------------------------------------------
                                  2                           2        2   
     (d, a*d), (d, a*b*d), (d, a*b d), (b, b), (b, b*d), (b, b ), (b, b d),
     ------------------------------------------------------------------------
                                  2          2                       2   
     (b, a*b), (b, a*b*d), (b, a*b ), (b, a*b d), (b*d, b*d), (b*d, b d),
     ------------------------------------------------------------------------
                            2      2   2     2   2      2     2     2 
     (b*d, a*b*d), (b*d, a*b d), (b , b ), (b , b d), (b , a*b ), (b ,
     ------------------------------------------------------------------------
        2      2    2      2      2                                   
     a*b d), (b d, b d), (b d, a*b d), (a, a), (a, a*d), (a, a*b), (a,
     ------------------------------------------------------------------------
                    2          2                                       2   
     a*b*d), (a, a*b ), (a, a*b d), (a*d, a*d), (a*d, a*b*d), (a*d, a*b d),
     ------------------------------------------------------------------------
                                        2            2                   
     (a*b, a*b), (a*b, a*b*d), (a*b, a*b ), (a*b, a*b d), (a*b*d, a*b*d),
     ------------------------------------------------------------------------
                2        2     2       2     2        2      2
     (a*b*d, a*b d), (a*b , a*b ), (a*b , a*b d), (a*b d, a*b d)}

o4 : List
i5 : C = coveringRelations P

                                                                 2      
o5 = {(1, d), (1, b), (1, a), (d, b*d), (d, a*d), (b, b*d), (b, b ), (b,
     ------------------------------------------------------------------------
                  2                    2   2      2     2     2      2       
     a*b), (b*d, b d), (b*d, a*b*d), (b , b d), (b , a*b ), (b d, a*b d), (a,
     ------------------------------------------------------------------------
                                                          2              2   
     a*d), (a, a*b), (a*d, a*b*d), (a*b, a*b*d), (a*b, a*b ), (a*b*d, a*b d),
     ------------------------------------------------------------------------
         2     2
     (a*b , a*b d)}

o5 : List

Ways to use coveringRelations :