Some frustrating questions on dimensions of products of posets
George M. Bergman
TL;DR
This work addresses how the dimension of a poset behaves under direct products, highlighting that additivity can fail in subtle ways and proposing a suite of auxiliary invariants to analyze this behavior. The central contribution is a general theorem giving upper bounds for $\dim(P\times\prod_{j\in n} C_j)$ under a coordinate-structure condition when $P$ embeds into a $d$-fold product, with several concrete corollaries showing dimension preservation or reduction in products such as $P\times C_0\times C_1$ and $P\times\mathbf{2}^2$. It also revisits a classical result: for finite-dimensional posets with both a least and a greatest element, $\dim(P\times Q)=\dim(P)+\dim(Q)$, and it surveys longstanding open questions about bounds on $\dim(P\times Q)$ and the effect of adding factors like $\mathbf{2}^n$, along with partial results and notable examples (e.g., $S_n$). The paper then develops and analyzes several auxiliary notions—absorbency, bounded dimension, and Boolean dimension—and examines their interaction with products, providing new bounds, equivalences, and open questions that offer fresh angles for tackling the broader problem of product-dimension in posets. Finally, it surveys other dimension notions (local, fractional) and extends the discussion to infinite posets via a compactness argument, illustrating how finite subposet behavior controls the infinite case. The cumulative aim is to map the landscape of product-dimension behavior, identify robust structural tools, and pose precise questions to guide future work in poset dimension theory.
Abstract
For $P$ a poset, the dimension of $P$ is defined to be the least cardinal $κ$ such that $P$ is embeddable in a direct product of $κ$ totally ordered sets. We study the behavior of this function on finite-dimensional (not necessarily finite) posets. In general, the dimension dim($P$ x $Q$) of a product of two posets can be smaller than dim($P$) + dim($Q$), though no cases are known where the discrepancy is greater than 2. We obtain a result that gives upper bounds on the dimensions of certain products of posets, including cases where the discrepancy 2 is achieved. But the paper is mainly devoted to stating questions, old and new, about dimensions of product posets, noting implications among their possible answers, and introducing some related concepts that might be helpful in tackling these questions.