# Example: Constructing common posets

The standard way to construct a Poset is the poset method. However, this package also provides many enumerators for common posets.

For example, we can construct a boolean lattice in many ways. First, we construct it with the booleanLattice method.

 i1 : n = 3; i2 : B = booleanLattice n;

We can also construct it as a product of length 2 chains.

 i3 : C2 = chain 2; i4 : C = product(n, i -> C2); i5 : areIsomorphic(B, C) o5 = true

Further, we can construct it as the divisorPoset of a product of primes.

 i6 : P = {2, 3, 5, 7, 11, 13, 17, 19}; i7 : D = divisorPoset product take(P, n); i8 : areIsomorphic(B, D) o8 = true

It is also the standardMonomialPoset of the Ideal of squares of the variables.

 i9 : R = QQ[x_1..x_n]; i10 : I = monomialIdeal apply(R_*, x -> x^2); o10 : MonomialIdeal of R i11 : M = standardMonomialPoset I; i12 : areIsomorphic(B, M) o12 = true

There are many other common posets that can be generated with this package. See the below list for the methods.

• booleanLattice -- generates the boolean lattice on $n$ elements
• chain -- generates the chain poset on $n$ elements
• divisorPoset -- generates the poset of divisors
• dominanceLattice -- generates the dominance lattice of partitions of $n$
• facePoset -- generates the face poset of a simplicial complex
• intersectionLattice -- generates the intersection lattice of a hyperplane arrangement
• lcmLattice -- generates the lattice of lcms in an ideal
• ncpLattice -- computes the non-crossing partition lattice of set-partitions of size $n$
• partitionLattice -- computes the lattice of set-partitions of size $n$
• plueckerPoset -- computes a poset associated to the Pluecker relations
• randomPoset -- generates a random poset with a given relation probability
• resolutionPoset -- generates a poset from a resolution
• standardMonomialPoset -- generates the poset of divisibility in the monomial basis of an ideal
• youngSubposet -- generates a subposet of Young's lattice