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# isSperner -- determines if a ranked poset has the Sperner property

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

## Description

The ranked poset $P$ is Sperner if the maximum size of a set of elements with the same rank is the dilworthNumber of $P$. That is, $P$ is Sperner if the maximum size of a set of elements with the same rank is the maximum size of an antichain.

The $n$ chain and the $n$ booleanLattice are Sperner.

    i1 : n = 5;

    i2 : isSperner chain n

    o2 = true

    i3 : isSperner booleanLattice n

    o3 = true

However, the following poset is non-Sperner as it has an antichain of size $4$ but the set of elements of rank $0$ and the set of elements of rank $1$ are both of size $3$.

    i4 : P = poset {{1,4}, {1,5}, {1,6}, {2,6}, {3,6}};

    i5 : isSperner P

    o5 = false

    i6 : isAntichain(P, {2,3,4,5})

    o6 = true

    i7 : rankGeneratingFunction P

    o7 = 3q + 3

    o7 : ZZ[q]

## See also

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

## Ways to use isSperner:

- `isSperner(Poset)`

## For the programmer

The object isSperner is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Posets.m2:6052:0.