poset -- creating a poset

Synopsis

Usage:
P = poset (G,R)
P = poset (G,R,M)
Inputs:
- G, a list, elements in the ground set of P
- R, a list, sequences (a,b) which indicate that a is less than or equal to b
- M, a matrix, with entries ij equal to one when the jth element in G is less than or equal to the ith element in G
Outputs:
- P, an object of class Poset, a poset consisting of the elements in G, with the order relations determined by R and or M

Description

This function allows one to create a poset by defining the set and giving the order relations between the elements in the set.

i1 : G = {a,b,c,d};

i2 : R = {(a,b), (a,c), (c,d)};

i3 : P = poset (G,R)

o3 = Poset{cache => CacheTable                  }
           GroundSet => {a, b, c, d}
           RelationMatrix => | 1 1 1 1 |
                             | 0 1 0 0 |
                             | 0 0 1 1 |
                             | 0 0 0 1 |
           Relations => {(a, b), (a, c), (c, d)}

o3 : Poset

Sometimes in finding "enough" relations one inadverdently finds all of the relations in the poset. In this case, the user can bypass having the function poset calculate the transitive closure of the relations by entering the matrix encoding the relations in when calling the poset function.

i4 : S = QQ[x,y,z];

i5 : G = {x^2, x*y, z^2, x^2*y*z, x*y*z^3, x^2*y^2*z^3};

i6 : R = select((flatten apply (G, g-> apply (G, h-> if h % g == 0 then (g,h)))), i -> i =!= null) -- finds all pairs where g divides h

        2   2     2   2        2   2 2 3                      2           
o6 = {(x , x ), (x , x y*z), (x , x y z ), (x*y, x*y), (x*y, x y*z), (x*y,
     ------------------------------------------------------------------------
          3          2 2 3     2   2     2       3     2   2 2 3     2    
     x*y*z ), (x*y, x y z ), (z , z ), (z , x*y*z ), (z , x y z ), (x y*z,
     ------------------------------------------------------------------------
      2        2      2 2 3         3       3         3   2 2 3     2 2 3 
     x y*z), (x y*z, x y z ), (x*y*z , x*y*z ), (x*y*z , x y z ), (x y z ,
     ------------------------------------------------------------------------
      2 2 3
     x y z )}

o6 : List

i7 : M = matrix apply (G, g-> apply (G, h-> if h % g == 0 then 1 else 0))

o7 = | 1 0 0 1 0 1 |
     | 0 1 0 1 1 1 |
     | 0 0 1 0 1 1 |
     | 0 0 0 1 0 1 |
     | 0 0 0 0 1 1 |
     | 0 0 0 0 0 1 |

              6        6
o7 : Matrix ZZ  <--- ZZ

i8 : P = poset(G,R,M)

o8 = Poset{cache => CacheTable                                                                                                                                                                                                                    }
                          2        2   2          3   2 2 3
           GroundSet => {x , x*y, z , x y*z, x*y*z , x y z }
           RelationMatrix => | 1 0 0 1 0 1 |
                             | 0 1 0 1 1 1 |
                             | 0 0 1 0 1 1 |
                             | 0 0 0 1 0 1 |
                             | 0 0 0 0 1 1 |
                             | 0 0 0 0 0 1 |
                           2   2     2   2        2   2 2 3                      2                 3          2 2 3     2   2     2       3     2   2 2 3     2      2        2      2 2 3         3       3         3   2 2 3     2 2 3   2 2 3
           Relations => {(x , x ), (x , x y*z), (x , x y z ), (x*y, x*y), (x*y, x y*z), (x*y, x*y*z ), (x*y, x y z ), (z , z ), (z , x*y*z ), (z , x y z ), (x y*z, x y*z), (x y*z, x y z ), (x*y*z , x*y*z ), (x*y*z , x y z ), (x y z , x y z )}

o8 : Poset

Ways to use `poset` :

poset(List,List)
poset(List,List,Matrix)

poset -- creating a poset

Synopsis

Description

See also

Ways to use poset :

Ways to use `poset` :