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Posets :: antichains

antichains -- computes all antichains of a poset

Synopsis

Description

A set of elements of P is called an antichain if no two distinct elements of the set are comparable.
i1 : D = divisorPoset 12;
i2 : antichains D

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

o2 : List
With the input k, the method restricts to only antichains of that length. In a divisorPoset, all chains of length 2 describe exactly the non-divisor-multiple pairs.
i3 : antichains(D, 2)

o3 = {{2, 3}, {3, 4}, {4, 6}}

o3 : List
Since every distinct pair of vertices in a chain is comparable, the only antichains of a chain are the singleton sets and the empty set.
i4 : antichains chain 5

o4 = {{}, {1}, {2}, {3}, {4}, {5}}

o4 : List

See also

Ways to use antichains :