Some frustrating questions on dimensions of products of posets

Abstract

The definition of the dimension of a poset is recalled. For a subposet P of a direct product of $d>0$ chains, and an integer $n>0$, a condition is developed which implies that for any family of n chains ${\left(T_j\right)}_{j\in n}$, one has $\dim (P\times {\prod }_{j\in n}{T}_j)\le d$. Applications are noted.

Open questions, old and new, on dimensions of product posets are stated, and some other numerical invariants of posets that seem useful for studying these questions are developed. Some variants of the concept of the dimension of a poset from the literature are also recalled.

In a final section, independent of the other results, it is noted that by the compactness theorem of first-order logic, an infinite poset P has finite dimension d if and only if d is the supremum of the dimensions of its finite subposets.