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chains -- computes all chains of a poset

Description

A set of elements of $P$ is called a chain if every pair of elements in the set are comparable.

i1 : D = divisorPoset 12;
 
i2 : chains D

o2 = {{}, {1}, {1, 2}, {1, 2, 4}, {1, 2, 4, 12}, {1, 2, 6}, {1, 2, 6, 12},
     ------------------------------------------------------------------------
     {1, 2, 12}, {1, 3}, {1, 3, 6}, {1, 3, 6, 12}, {1, 3, 12}, {1, 4}, {1, 4,
     ------------------------------------------------------------------------
     12}, {1, 6}, {1, 6, 12}, {1, 12}, {2}, {2, 4}, {2, 4, 12}, {2, 6}, {2,
     ------------------------------------------------------------------------
     6, 12}, {2, 12}, {3}, {3, 6}, {3, 6, 12}, {3, 12}, {4}, {4, 12}, {6},
     ------------------------------------------------------------------------
     {6, 12}, {12}}

o2 : List

With the input k, the method restricts to only chains of that length. In a divisorPoset, all chains of length $2$ describe exactly the divisor-multiple pairs.

i3 : chains(D, 2)

o3 = {{1, 2}, {1, 3}, {1, 4}, {1, 6}, {1, 12}, {2, 4}, {2, 6}, {2, 12}, {3,
     ------------------------------------------------------------------------
     6}, {3, 12}, {4, 12}, {6, 12}}

o3 : List

See also

Ways to use chains:

  • chains(Poset)
  • chains(Poset,ZZ)

For the programmer

The object chains is a method function.

The source of this document is in Posets.m2:4724:0.