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 $b \leq d$.

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.

