Weakly o-minimal structure

In model theory, a weakly o-minimal structure is a model-theoretic structure whose definable sets in the domain are just finite unions of convex sets.

Version

v1 · 2026-09-28 · History

Domain-specific #

12852

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomains

Model Theory, O Minimality → Mathematics

Core Idea

Weakly o-minimal structure is treated here as the recurring mathematicslogicstatistics identity summarized by this source-grounded definition: In model theory, a weakly o-minimal structure is a model-theoretic structure whose definable sets in the domain are just finite unions of convex sets. In model theory, a weakly o-minimal structure is a model-theoretic structure whose definable sets in the domain are just finite unions of convex sets. For weakly o-minimal structures (M, this is relaxed so that the definable sets in M are finite unions of convex definable sets.

Scope of Application

• Definition. For weakly o-minimal structures (M, this is relaxed so that the definable sets in M are finite unions of convex definable sets.

• Definition. A set C is convex if whenever a and b are in C , a M satisfies that a 0.

• Definition. so that the set consists of all strictly positive real algebraic numbers that are less than π.

• Definition. The set is clearly convex, but cannot be written as a finite union of points and intervals whose endpoints are in R.

• Definition. To write it as an interval one would either have to include the endpoint π, which isn't in R, or one would require infinitely many intervals, such as the union.

Clarity

A clear use of Weakly o-minimal structure names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In model theory, a weakly o-minimal structure is a model-theoretic structure whose definable sets in the domain are just finite unions of convex sets.

Manages Complexity

Weakly o-minimal structure compresses multiple mathematicslogicstatistics details into a stable diagnostic relation. The source shows both the central mechanism—for weakly o-minimal structures (M, this is relaxed so that the definable sets in M are finite unions of convex definable sets.—and the practical consequence—since we have a definable set that isn't a finite union of points and intervals, this structure is not o-minimal.

Abstract Reasoning

1. Type the carrier. Identify the mathematicslogicstatistics entities to which the claim applies.
2. State the relation. Use the source-grounded identity: In model theory, a weakly o-minimal structure is a model-theoretic structure whose definable sets in the domain are just finite unions of convex sets.
3. Check operation and conditions. A set C is convex if whenever a and b are in C , a M satisfies that a 0.
4. Demand recognition evidence.

Knowledge Transfer

Within the home domain. Knowledge about Weakly o-minimal structure transfers literally when a new case preserves the same carrier type, relation, and recognition test. For weakly o-minimal structures (M, this is relaxed so that the definable sets in M are finite unions of convex definable sets. A set C is convex if whenever a and b are in C , a M satisfies that a 0. Beyond the home domain. No canonical parent is asserted for Weakly o-minimal structure.

Neighborhood in Abstraction Space

Weakly o-minimal structure sits in a moderately populated region (50^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Geometric Figures & Constructions (32 abstractions)

Nearest neighbors

• T4 Space — 0.86
• Hereditarily normal space — 0.86
• Indiscrete space — 0.86
• Julia set — 0.86
• Filling radius — 0.86

Computed from structural-signature embeddings · 2026-10-08