Definably Complete Locally O-Minimal Expansions

• Definably complete locally o-minimal expansions are ordered structures where every bounded definable set has a supremum and local o-minimality ensures simple local behavior.
• They support robust dimension theory and cell-decomposition, yielding results like the dimension inequality for definable maps and a definable Baire category theorem.
• DCULOAS frameworks bridge classical o-minimal geometry with broader tame systems, offering applications in real analysis, geometry, and model theory.

A definably complete locally o-minimal expansion is an ordered structure with strong topological and combinatorial regularity but weaker than full o-minimality. The subclass of definably complete uniformly locally o-minimal expansions of the second kind—often abbreviated in the literature as DCULOAS—exhibits a uniform local geometric tameness governed by a two-parameter cell-slicing property, combining uniform local o-minimality with definable completeness. This framework permits a robust theory of dimension and cell-decomposition, supporting results parallel to those in o-minimal structures, such as the dimension inequality for definable maps and a definable Baire category theorem. These expansions serve as a bridge between classical o-minimal geometry and broader model-theoretic “tame” systems, with applications in geometry, analysis, and model theory.

1. Foundational Concepts and Definitions

A definably complete locally o-minimal expansion is a first-order structure $\mathcal{R}=(R,<,+,0,\dots)$, where $(R,<,+,0)$ is a densely ordered abelian group (often, but not necessarily, an ordered field), such that:

• Definable completeness: Every definable (possibly with parameters) subset of $R$ that is bounded above has a least upper bound in $R$. Equivalently, for every definable $X \subset R$ nonempty and bounded above, $\sup X\in R$ (Fujita, 2020).
• Local o-minimality: For every definable $X\subset R$ and every $a\in R$, there exists an open interval $I\ni a$ such that $X\cap I$ is a finite union of points and open intervals (Fujita, 2020).

A uniformly locally o-minimal expansion of the second kind strengthens this by requiring a two-parameter uniform cell decomposition, so that on every small box of radius $r$ one can decompose definable sets into finitely many simple components of size controlled by $s$—formalized as families $\{ X_{r,s} \}_{r,s>0}$ of closed, bounded, definable sets with appropriate monotonicity and nesting properties (Fujita, 2020, Fujita, 2020).

2. Structural Properties and Dimension Theory

In a DCULOAS structure, a robust theory of definable dimension emerges, capturing “topological size” in analogy with o-minimal geometry. The dimension $\dim(X)$ of a definable $X\subset R^n$ is:

$$\dim(X) = \max\left\{ k : X \text{ projects onto a nonempty open subset of } R^k \right\}$$

This dimension notion satisfies key properties:

• Monotonicity: $X\subset Y$ implies $\dim X \leq \dim Y$
• Additivity: $\dim(X \times Y) = \dim X + \dim Y$
• Finite Union: $\dim(X \cup Y) = \max( \dim X, \dim Y )$
• Frontier: The boundary of a nonempty $X$ has strictly smaller dimension, i.e., $\dim(\partial X)<\dim(X)$
• Projection: For definable $f:X\to R^n$, $\dim(f(X))\leq\dim X$ (Fujita, 2020, Fujita, 2022, Fujita, 2020)

An important result is the dimension-additivity property for surjective definable maps with equi-dimensional fibers: if $f:X\to Y$ is surjective and all fibers have the same dimension $d$, then $\dim X = \dim Y + d$ (Fujita, 2020).

3. Main Results: Dimension Inequality and the Definable Baire Property

A central theorem for DCULOAS is the dimension inequality for definable maps:

• If $f: X \rightarrow R^n$ is any definable map, then

$\dim(f(X)) \leq \dim(X)$

(Fujita, 2020).

The proof leverages uniformity in local cell decomposition, definable completeness for selection arguments, and the definably Baire property. The induction proceeds by reducing to open boxes of full dimension and analyzing fibers via graphs and projections, ultimately constructing definable continuous sections by lexicographic minima.

A powerful corollary is that the set of points where a definable function is discontinuous has strictly lower dimension than its domain, enforcing strong regularity of definable functions (Fujita, 2020, Fujita, 2020).

Additionally, Fujita showed that such structures are definably Baire: no increasing definable union of nowhere-dense sets fills an open set, paralleling the classical Baire category theorem (Fujita, 2020).

4. Pregeometry, Rank, and Comparison to the O-Minimal Setting

DCULOAS expansions induce a pregeometry (matroid) structure via discrete closure: the closure operator “discl,” where $x\in \operatorname{discl}(A)$ if $x$ belongs to some $A$-definable closed-discrete set. This closure operation yields a well-behaved rank function:

$$\operatorname{rk}_{\operatorname{discl}}(X|A) = \dim(X)$$

There is a three-way coincidence between matroid rank, projection dimension, and first-order topological dimension (Fujita, 2022). In the o-minimal context, the closure operation collapses to algebraic (or definable) closure, and dimension/rank theory aligns precisely with classic o-minimal results.

5. Decomposition and Tame Geometric Structures

DCULOAS structures admit (weak) cell-like decompositions. Every definable set can be partitioned into special submanifolds (or quasi-special submanifolds), locally given as graphs of continuous definable maps over open boxes, and equipped with definable tubular neighborhoods (Fujita, 2022, Fujita, 2020). This decomposition is compatible with the “frontier condition” (closure of each piece is a union of pieces) and refines every finite family of definable sets.

These properties enable the transfer of much of o-minimal tame geometry to contexts with only local cell structure and definable completeness, despite potential global pathologies (e.g., infinite discrete sets).

6. Examples and Applications

Concrete examples of DCULOAS include:

• Expansions of $(\mathbb{R},+,<)$ by infinite discrete subgroups or restricted analytic functions (subject to uniform growth conditions) (Fujita, 2020).
• Structures where $R$ is the union of $(R, <, +, 0)$ with unary predicates for $\mathbb{Q}$ or other discrete subsets, under suitable tameness (Fujita, 2020).
• Ultraproducts of o-minimal expansions exhibiting only local, not global, o-minimality (Fujita et al., 2023).

Applications span tame topology (definable versions of invariance of domain (Fujita, 26 Jan 2026)), functional analysis (definable Arzelà–Ascoli and Tietze extension theorems (Fujita, 2020)), and quotient constructions for definable group actions (Fujita et al., 2022).

7. Significance and Further Directions

Definably complete locally o-minimal expansions, especially those uniformly locally o-minimal of the second kind, provide a foundational model-theoretic setting for the study of “tame” structures beyond o-minimality. They generalize cell decomposition, dimension theory, and Baire-category methods to ordered environments lacking global field structure or full o-minimality. The significance lies in preserving topological and combinatorial regularity sufficient for geometry, real analysis, and model theory, while broadening the landscape of examples and behaviors. Open questions concern the limits of cell decomposition, tameness criteria for expansions, and the development of further geometric invariants such as Euler characteristics in this setting (Fujita, 2020, Fujita, 2022, Fujita, 2020).