Further thoughts on dimensions of posets

We recall the concept of the dimension of a finite poset $P$, and the longstanding conjecture that for all finite nonempty posets $P$ and $Q$, $\dim (P\times Q)\ge \dim (P)+\dim (Q)-2.$ We then note two other plausible inequalities, either of which would imply that one. In the final section, writing $P\preccurlyeq P^{\mathrm{\prime }}$ if, for all $Q,$ $\dim (P\times Q)\le \dim (P^{\mathrm{\prime }}\times Q),$ and writing $P\approx P^{\mathrm{\prime }}$ if $P\preccurlyeq P^{\mathrm{\prime }}$ and $P^{\mathrm{\prime }}\preccurlyeq P,$ we note some results and questions concerning these relations.