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

- 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

## See also

- diamondProduct -- computes the diamond product of two ranked posets

## Ways to use this method:

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

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Posets.m2:3231:0.