product(Poset,Poset) -- computes the product of two posets

Synopsis

Function: product
Usage:

R = product(P, Q)

R = P * Q
Inputs:
- P, an instance of the type Poset,
- Q, an instance of the type Poset,
Outputs:
- R, an instance of the type Poset, the Cartesian product of $P$ and $Q$

Description

The Cartesian product of the posets $P$ and $Q$ is the new poset whose ground set is the Cartesian product of the ground sets of $P$ and $Q$ and with partial order given by $(a,b) \leq (c,d)$ if and only if $a \leq c$ and $b \leq d$.

i1 : product(chain 3, poset {{a,b},{b,c}})

o1 = Relation Matrix: | 1 1 1 1 1 1 1 1 1 |
                      | 0 1 1 0 1 1 0 1 1 |
                      | 0 0 1 0 0 1 0 0 1 |
                      | 0 0 0 1 1 1 1 1 1 |
                      | 0 0 0 0 1 1 0 1 1 |
                      | 0 0 0 0 0 1 0 0 1 |
                      | 0 0 0 0 0 0 1 1 1 |
                      | 0 0 0 0 0 0 0 1 1 |
                      | 0 0 0 0 0 0 0 0 1 |

o1 : Poset

The product of $n$ chains of length $2$ is isomorphic to the boolean lattice on $n$ elements. These are also isomorphic to the divisor lattice on the product of $n$ distinct primes.

i2 : B = booleanLattice 4;

i3 : B == product(4, i -> chain 2)

o3 = true

Ways to use this method: