naturalLabeling -- relabels a poset with a natural labeling

Synopsis

Usage:
Q = naturalLabeling P
Q = naturalLabeling(P, startIndex)
Inputs:
- P, an object of class Poset
- startIndex, an integer, the starting index from which the poset is relabeled (default 0)
Outputs:
- Q, an object of class Poset

Description

A poset is naturally labeled if the ground set is ordered v₁, ..., v_n and if v_i ≤v_j in P implies i ≤j. This method relabels the ground set of the poset (suppose it has n vertices) to be 0, 1, ..., n-1.

i1 : P = booleanLattice 3;

i2 : Q = naturalLabeling P

o2 = Poset{cache => CacheTable{...13...}                                                                                }
           GroundSet => {0, 1, 2, 4, 3, 5, 6, 7}
           RelationMatrix => | 1 1 1 1 1 1 1 1 |
                             | 0 1 0 1 0 1 0 1 |
                             | 0 0 1 1 0 0 1 1 |
                             | 0 0 0 1 0 0 0 1 |
                             | 0 0 0 0 1 1 1 1 |
                             | 0 0 0 0 0 1 0 1 |
                             | 0 0 0 0 0 0 1 1 |
                             | 0 0 0 0 0 0 0 1 |
           Relations => {{0, 1}, {2, 4}, {0, 2}, {1, 4}, {3, 5}, {6, 7}, {3, 6}, {5, 7}, {0, 3}, {1, 5}, {2, 6}, {4, 7}}

o2 : Poset

i3 : all(allRelations Q, r -> r_0 <= r_1)

o3 = true

If startIndex is specified, then the values are shifted by that amount. This can be useful for making a disjoint union of posets.

i4 : C = chain 3;

i5 : Q' = sum(3, i -> naturalLabeling(C, 3*i))

o5 = Poset{cache => CacheTable{}                                        }
           GroundSet => {0, 1, 2, 3, 4, 5, 6, 7, 8}
           RelationMatrix => | 1 1 1 0 0 0 0 0 0 |
                             | 0 1 1 0 0 0 0 0 0 |
                             | 0 0 1 0 0 0 0 0 0 |
                             | 0 0 0 1 1 1 0 0 0 |
                             | 0 0 0 0 1 1 0 0 0 |
                             | 0 0 0 0 0 1 0 0 0 |
                             | 0 0 0 0 0 0 1 1 1 |
                             | 0 0 0 0 0 0 0 1 1 |
                             | 0 0 0 0 0 0 0 0 1 |
           Relations => {{0, 1}, {1, 2}, {3, 4}, {4, 5}, {6, 7}, {7, 8}}

o5 : Poset

i6 : all(allRelations Q', r -> r_0 <= r_1)

o6 = true

Note the cache of P is copied to the cache of Q with the appropriate adjustments being made.

Ways to use `naturalLabeling` :

naturalLabeling(Poset)
naturalLabeling(Poset,ZZ)

naturalLabeling -- relabels a poset with a natural labeling

Synopsis

Description

See also

Ways to use naturalLabeling :

Ways to use `naturalLabeling` :