Finite products of pruned trees and dense matrices

A tree here has height $\omega$, a unique root, finitely many immediate successors at each node, and no terminal nodes. Heights, tree order and levels are as in Set-theoretic trees, heights, levels, branches and antichains. For $A\subseteq T$, say that $A$ dominates $t$ if $t{\le }_{T}a$ for some $a\in A$. For $h,k<\omega$, $A$ is $(h,k)$-dense if some $x\in T_h$ has every node of $T_{h+k}$ above $x$ dominated by $A$. It is $k$-dense if it is $(0,k)$-dense, and infinity-dense if it is $k$-dense for every $k<\omega$.

The unique root is below every node: the first predecessor of a positive-height node is a root, and uniqueness identifies it; the height-zero case is the root itself. Hence the height-$k$ cone above the root is exactly $T_k$. Therefore $A$ is $k$-dense iff it dominates every node on level $k$, in both directions by this equality. Every node has some finite height, so infinity-density implies it is dominated by applying this equivalence at its height. Conversely if every node is dominated, then every level is dominated and the same equivalence gives $k$-density for each $k$. These prove both density characterizations directly.

For a positive finite family $(T_1,\dots ,T_d)$ of such trees, an $(h,k)$-matrix is a product ${\prod }_{i=1}^d{A}_i$ where each $A_i\subseteq T_i$ is $(h,k)$-dense; the same $h,k$ are used in every factor. A $k$-matrix means a $(0,k)$-matrix. Matrices are subsets of the full product ${\prod }_i{T}_i$. The level product, in contrast, is ${\bigcup }_{n<\omega }{\prod }_i(T_i{)}_n$, consisting only of equal-height tuples. The density definition does not require a matrix to be in the level product.

For $k=0$, $(h,0)$-density means that some node of level $h$ is dominated; for $h=k=0$, it is equivalent to $A\mathrm{\ne }\mathrm{\varnothing }$. No terminal nodes ensures every height-$h$ node has an extension at height $h+k$, by finitely many successor choices; therefore an empty set is never $(h,k)$-dense. At $d=1$ a matrix is just the indicated dense set, up to the one-tuple identification; $d=0$ is excluded.