Further thoughts on dimensions of posets

Abstract

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 $P\preccurlyeq P^{\prime }$ if, for all $Q,$ $\dim (P\times Q)\le \dim (P^{\prime }\times Q),$ and writing $P\approx P^{\prime }$ if $P\preccurlyeq P^{\prime }$ and $P^{\prime }\preccurlyeq P,$ we note some results and questions concerning these relations.

This is an addendum to my paper at arXiv:2312.12615. 4 pages. Copy at https://math.berkeley.edu/~gbergman/papers/unpub/ or if I decide to publish it, https://math.berkeley.edu/~gbergman/papers/ , may be updated more frequently than arXiv copy