## Franklin's Notes

### Product of lattices

Given two lattices $L_1$ and $L_2$, their (direct) product, denoted $L_1\times L_2$, is a lattice whose underlying set is the cartesian product of $L_1$ and $L_2$, and whose ordering is defined by letting $(\ell_1,\ell_2)\leq (\ell'_1,\ell'_2)$ if and only if $\ell_1\leq \ell'_1$ and $\ell_2\leq\ell'_2$. For instance, we have the following lattice product:

If a lattice $L$ is isomorphic to $L_1\times L_2$ for two other (nonsingleton) lattices $L_1,L_2$, then it is called composite, and otherwise it is called prime.