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filtration -- generates the filtration of a poset

Synopsis

Description

The filtration of $P$ is a partitioning $F$ of the vertices such that $F_0$ is the set of minimal elements of $P$, $F_1$ is the set of minimal elements of $P - F_0$, 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 :

For the programmer

The object filtration is a method function.