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Posets :: isStrictSperner

isStrictSperner -- determines if a ranked poset has the strict Sperner property

Synopsis

Usage:

i = isStrictSperner P
Inputs:
- P, an instance of the type Poset, a ranked poset
Outputs:
- i, a Boolean value, whether $P$ is strict Sperner

Description

The ranked poset $P$ is strict Sperner if the maximalAntichains are the sets of elements with the same rank.

The $n$ chain is strict Sperner as the maximal antichains and the sets of elements with the same rank are singletons.

i1 : isStrictSperner chain 5

o1 = true

The $n$ booleanLattice, for $n \geq 3$, is not strict Sperner as it has maximal antichains which are not sets of elements with the same rank.

i2 : B = booleanLattice 3;

i3 : isStrictSperner B

o3 = false

i4 : rankPoset B

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

o4 : List

i5 : maximalAntichains B

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

o5 : List

See also

dilworthNumber -- computes the Dilworth number of a poset
isRanked -- determines if a poset is ranked
isSperner -- determines if a ranked poset has the Sperner property
maximalAntichains -- computes all maximal antichains of a poset
rankFunction -- computes the rank function of a ranked poset

Ways to use `isStrictSperner` :

"isStrictSperner(Poset)"

For the programmer

The object isStrictSperner is a method function.