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
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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 |