Definable Ranks of Ordered Fields

• The paper demonstrates that every pattern of definable convex end-segments in linear orders is realized as the definable rank of some ordered field.
• It employs lexicographic constructions and natural valuations to establish canonical isomorphisms among the ranks of the field, its value group, and its archimedean spine.
• The study extends to almost real closed fields, showing that definable ranks coincide across field, value group, and value set, which paves the way for further research in valuation theory.

Definable ranks of ordered fields constitute a model-theoretic framework capturing the hierarchy of convex substructures that are definable in the first-order language of ordered fields. This invariant refines the classical notion of rank, which classifies convex valuation rings (or subgroups, or segments), by restricting attention to those that admit a first-order definition. Recent advances have sharply clarified the structure and flexibility of definable ranks, demonstrating that all patterns of definable end-segments observed for linear orders arise as definable ranks in the setting of ordered fields, thereby fully characterizing the model-theoretic complexity that ordered fields can encode in their convex valuation theory (Boissonneau et al., 20 Dec 2025). A parallel theory holds in the context of almost real closed fields, where definable rank coincidences across field, value group, and value set are delineated (Krapp et al., 31 May 2025).

1. Model-Theoretic Notion of Definable Rank

In the language $\mathcal L$ of ordered structures, rank invariants are defined as follows:

• For a linear order $(\Gamma,<)$, the rank $\mathrm{rk}(\Gamma)$ is the set of all proper end-segments, ordered by inclusion. The definable rank $\mathrm{drk}(\Gamma)$ comprises those end-segments first-order definable in $(\Gamma,<)$.
• For an ordered abelian group $(G,+,<)$, $\mathrm{rk}(G)$ is the set of all proper convex subgroups, and $\mathrm{drk}(G)$ is the sub-poset of those convex subgroups definable in $\{0,+,-,<\}$.
• For an ordered field $(K,+,\cdot,<)$, $\mathrm{rk}(K)$ is the collection of proper convex valuation rings $\mathcal O\subsetneq K$, and $\mathrm{drk}(K)$ those rings that are definable (with parameters) in $\{0,1,+,\cdot,<\}$ (Boissonneau et al., 20 Dec 2025, Krapp et al., 31 May 2025).

The process thus yields three interrelated hierarchies for any ordered field: on the underlying field, its canonical value group, and its archimedean spine. These structures admit canonical order-preserving maps, and in the purely set-theoretic (i.e., abstract, non-definable) case, are isomorphic.

2. Archimedean Spine and Natural Valuation: Structural Intermediaries

The archimedean spine of an ordered abelian group $(G,<)$ is the linearly ordered quotient $\Gamma = G/\sim$, where $x\sim y$ if there exist $n,m\in\mathbb N$ such that $n|x|\ge|y|$ and $m|y|\ge|x|$. The ordering on classes $[x]>[y]$ corresponds to $|x|<|y|$ and $x\not\sim y$ (Boissonneau et al., 20 Dec 2025).

For ordered fields, the natural valuation $v_{\mathrm{nat}}$ on $K$ arises by quotienting $K^\times$ by the same archimedean equivalence, producing a value group $G=K^\times/\sim$ and residue field archimedean (a subfield of $\mathbb R$). The value set $\Gamma$ is again the archimedean spine of $G$, tightly connecting the three layers of structure (Krapp et al., 31 May 2025).

3. Isomorphism Theorems and Main Constructions

The principal result is as follows: Given any linear order $(\Gamma,<)$, there exists an ordered abelian group $G$ with archimedean spine $\Gamma$ and an ordered field $K$ whose natural value group is $G$, such that

$$\mathrm{drk}(\Gamma) \cong \mathrm{drk}(G) \cong \mathrm{drk}(K).$$

This holds universally: every pattern of definable convex end-segments that appears in a linear order is instantiated as the definable rank of some ordered field (Boissonneau et al., 20 Dec 2025).

Construction Scheme

• From $\Gamma$ to $G$: Employ $$\displaystyle G = \bigoplus_{i \in \Gamma} \mathbb Z$$ (the lexicographically ordered direct sum of copies of $\mathbb Z$ indexed by $\Gamma$). Convex subgroups correspond bijectively to sums over end-segments $\Delta\subseteq\Gamma$; the definable ones to definable end-segments; hence $\mathrm{drk}(G) \cong \mathrm{drk}(\Gamma)$.
• From $G$ to $K$: Form $K = \mathbb R((G))$, the Hahn series field with real coefficients and value group $G$. The natural valuation $v_{\mathrm{nat}}$ is henselian and definable (as no nontrivial $\Delta$ can simultaneously force both required alignments for non-definability), hence the induced map $\Phi_K: \mathrm{drk}(K)\to \mathrm{drk}(G)$ is an isomorphism (Boissonneau et al., 20 Dec 2025).

4. Definable Rank Coincidence in Almost Real Closed Fields

In almost real closed fields—those real fields whose natural valuation is henselian with real-closed residue field—there is a full correspondence among definable ranks on the field, its value group, and the associated value set. Explicitly,

$$(\mathrm{drk}_K, \subseteq) \cong (\mathrm{drk}_G, \subseteq) \cong (\mathrm{drk}_\Gamma, \subseteq).$$

This result leverages both algebraic properties (henselianity and residue closure) and model-theoretic analyses of definability, corresponding to Krapp–Kuhlmann–Vogel's theorem (Krapp et al., 31 May 2025).

Definable Rank Structure in Examples

For discrete $\Gamma$, every definable final segment has a least element and

$$\mathrm{drk}_\Gamma \cong \Gamma + 1 \quad \text{or} \quad 1+\Gamma+1.$$

For dense $\Gamma$ with endpoints,

$$\mathrm{drk}_\Gamma \cong \sum_{\gamma\in\Gamma} \{1,2\},$$

subject to extensions if endpoints exist (Krapp et al., 31 May 2025).

5. Classification, Co-augmentability, and Open Problems

Classifying possible definable ranks of ordered fields reduces to classifying which families of definable end-segments $\mathcal C$ arise from some linear order $\Gamma$. By the co-augmentability criterion: an end-segment $I\subseteq \Gamma$ is $\mathcal L_{<}$-definable if and only if the pair $(\Gamma\setminus I, I)$ is not co-augmentable (cf. Proposition 2.27 in [itme‐spines]; (Boissonneau et al., 20 Dec 2025)). This gives a complete description but is often highly nonexplicit; determining co-augmentability may be as subtle as hard model-theoretic or number-theoretic problems.

A consequence is that every definable rank structure realized in linear orders also appears for some ordered field. Conversely, to decide if a chain $C$ occurs as $\mathrm{drk}(K)$ requires building $\Gamma$ with definable end-segments exactly $C$. The open-ended nature of co-augmentability ensures the classification problem remains intricate.

6. Variability and Mismatches in Definable Ranks

Concrete constructions showcase the versatility and possible discrepancies in definable rank structures. For instance, given a non-definable end-segment $\Delta$ in $\Gamma$, one may define

$$G = \bigoplus_{\Gamma\setminus\Delta} \mathbb Z_{(2)} \oplus \bigoplus_\Delta \mathbb Z_{(3)}, \quad K = \mathbb R((G)),$$

yielding $\mathrm{drk}(G)$ that strictly exceeds $\mathrm{drk}(\Gamma)$. The universal construction (lexicographic sum by $\mathbb Z$) always guarantees equality of definable ranks. This flexibility supports the realization of all possible “mismatches” among $\mathrm{drk}(\Gamma)$, $\mathrm{drk}(G)$, and $\mathrm{drk}(K)$ (Boissonneau et al., 20 Dec 2025).

A notable example distinguishes all three structures: setting $$k=\mathbb R(t^{1/n}: n\in\mathbb N)\subsetneq\mathbb R((\mathbb Q))$$ and $K=k((\mathbb Q\sqcup\mathbb Q))$, then $\mathrm{drk}_K$ is a singleton, $\mathrm{drk}_G$ is empty, and $\mathrm{drk}_\Gamma$ corresponds to the two final segments of a three-point chain (Krapp et al., 31 May 2025).

7. Corollaries and Directions for Further Research

Key corollaries include:

• In almost real closed fields, definable convex $p$-coarsenings account for all definable valuations. If $\Gamma$ is well-ordered or reverse-well-ordered, then all abstract ranks are definable and $\mathrm{drk}_\Gamma = \mathrm{rk}_\Gamma$ (Krapp et al., 31 May 2025).
• Open questions: Which triples $(A,B,C)$ of order types can occur as $$(\mathrm{drk}_K,\mathrm{drk}_G,\mathrm{drk}_\Gamma)$$? What tameness conditions enforce $\mathrm{drk}_K \cong \mathrm{drk}_G$ beyond almost real closed fields? Which final segments of a linear order are $\mathcal L_{<}$-definable? These address fundamental interfaces between valuation theory, order theory, and model theory, and remain central to ongoing research (Boissonneau et al., 20 Dec 2025, Krapp et al., 31 May 2025).