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

- 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.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Posets.m2:6092:0.