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Posets :: filtration

filtration -- generates the filtration of a posett

Synopsis

Description

The filtration of P is a partitioning F of the vertices such that F0 is the set of minimal elements of P, F1 is the set of minimal elements of P - F0, and so forth.
i1 : P = poset {{a,b}, {b,c}, {c,d}, {a,e}, {e,d}};
i2 : filtration P

o2 = {{a}, {e, b}, {c}, {d}}

o2 : List
The filtration of a ranked poset is the same as the ranking of the poset.
i3 : B = booleanLattice 3;
i4 : F = filtration B

o4 = {{000}, {001, 010, 100}, {011, 101, 110}, {111}}

o4 : List
i5 : R = rankPoset B

o5 = {{000}, {001, 010, 100}, {011, 101, 110}, {111}}

o5 : List
i6 : sort \ F === sort \ R

o6 = true
The flatten of the filtration is a linear extension of the poset.
i7 : member(flatten F, linearExtensions B)

o7 = true
This method was ported from John Stembridge’s Maple package available at http://www.math.lsa.umich.edu/~jrs/maple.html#posets.

See also

Ways to use filtration :