I'm trying to characterize the join-irreducible elements of the direct product of two lattices $L$ and $K$.
If we denote by $\mathscr{J}(L)$ the set of all join-irreducible elements of $L$, then my guess so far is that $(a,b) \in \mathscr{J}(L \times K)$ iff $a \in \mathscr{J}(L)$ or (exclusive) $b \in \mathscr{J}(K)$. I've tried to prove this, but I'm getting stuck with some technical details. Moreover, my guess looks right when you look at diagrams of finite lattices.
Can someone give me a hand with this conjecture?