Let $P,Q$ be partially ordered sets such that $P\times Q$ is self dual. Self dual means $(P\times Q)^*=P\times Q$ where $P^*$ is the dual of $P$. Does that mean $P,Q$ are self dual?
My professor hinted that the answer is negative, that in particular there exist non self dual $P,Q: P\times Q$ is self dual. Why is that true? Is there a counterexample of two such sets?
Consider the partial orders $\langle\Bbb N,\le\rangle$ and $\langle\Bbb N,\ge\rangle$. Neither is self-dual — each is the dual of the other, in fact — but their product is self-dual.