Weakly binary expansions of dense meet-trees

TL;DR

This paper analyzes domination monoids of invariant types in the dense meet-tree theory $\mathsf{DMT}$ and its binary cone-expansions, such as $\mathsf{DTR}$. It first proves weak binarity for $\mathsf{DMT}$ and its cone-expansions, ensuring the domination monoid $\widetilde{Inv}(\mathfrak U)$ is well-defined. In the pure case, it shows $\widetilde{Inv}(\mathfrak U) \cong {\mathscr P}_{\mathrm{fin}}(X) \oplus \bigoplus_{g\in \mathfrak U} \mathbb N$, where $X$ is the set of grafts and the $\mathbb N$-summands arise from invariant $1$-types in new open cones above points. For purely binary cone-expansions, the monoid generalizes to $\widetilde{Inv}(\mathfrak U) \cong {\mathscr P}_{\mathrm{fin}}(X) \oplus \bigoplus_{g\in \mathfrak U} I_g$, with $I_g$ the monoid of the induced open-cone structure on $O_g$, providing a modular decomposition that applies to $\mathsf{DTR}$ and related theories. Overall, the work connects valuation-theoretic motivators to model-theoretic domination, yielding explicit, decomposed descriptions of domination monoids in dp-minimal expansions of dense trees.

Abstract

We compute the domination monoid in the theory DMT of dense meet-trees. In order to show that this monoid is well-defined, we prove weak binarity of DMT and, more generally, of certain expansions of it by binary relations on sets of open cones, a special case being the theory DTR from arXiv:1909.04626. We then describe the domination monoids of such expansions in terms of those of the expanding relations.

Figures (4)

• Figure 1: the point $a$ is in the same open cone above $g$ as the point $b$, while $c$ is in a different open cone above $g$.
• Figure 2: some nonrealised $B$-invariant types, where points of $B$ are denoted by triangles. In this picture, the set of triangles below $x$ has no maximum, solid lines lie in $\mathfrak U$, and dotted lines lie in a bigger $\mathfrak U_1\mathrel{^+\!\!\succ} \mathfrak U$. The type of $x$ is of kind (Ib), that of $y$ of kind (II), and that of $z$ of kind (IIIb).
• Figure 3: how to choose $a_i$ in the proof of Lemma \ref{['lemma:meetclosedmodulofinite3']}. In the first three pictures, $C^\mathfrak U_{b_i}$ has a maximum, $g$, denoted by a triangle. In the last picture it does not have one. Solid lines lie in $\mathfrak U$, and dotted lines lie in a bigger $\mathfrak U_1\mathrel{^+\!\!\succ} \mathfrak U$.
• Figure 5: proof of Proposition \ref{['pr:treebasic']}, how to show that $q(y)\mathrel{\ge_\mathrm{D}} p(x)$. In this picture $A$ only contains the point $c$, denoted by a triangle. Solid lines lie in $\mathfrak U$, and dotted lines lie in a bigger $\mathfrak U_1\mathrel{^+\!\!\succ} \mathfrak U$.

Theorems & Definitions (47)

• Theorem A: Theorem \ref{['thm:expdmt']}
• Theorem B: Theorem \ref{['thm:dmt']}
• Theorem C: Theorem \ref{['thm:invtildexp']}
• Definition 1.1
• Remark 1.3
• Definition 1.4
• Definition 1.5
• Lemma 1.6
• proof
• Definition 2.1