C-minimal theory

A model-theoretic theory whose definable one-variable sets are finite Boolean combinations of cones determined by a ternary C-relation.

Version

v2 · 2026-09-19 · History

Domain-specific #

3566

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomain

Model Theory → Mathematics

Core Idea

Definitions depend on the C-relation axioms and elementary extensions, and valued fields supply major examples analogous to o-minimal ordered structures. The ternary relation organizes a tree-like betweenness geometry, and minimality restricts definable subsets to simple cone configurations in every model. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

The load-bearing residual is not the broad topic of model theory. It is the domain-specific identity fixed by the language and complete theory, ternary C-relation and axioms, models and elementary extensions, cone or ball definitions, one-variable definable-set condition and examples such as algebraically closed valued fields are explicit.

How would you explain it like I'm…

 

No faithful explanation at this level. Two of three generators judged it unreachable: a child-level picture reduces it to one branching tree whose describable groups are whole branches, collapsing a property of a formal theory across all its models into a property of a single picture.

Tidy Tree-World Rules

Mathematicians sometimes study things arranged like the branches of a tree, using a special rule that looks at three points at a time and says which two are on a branch that splits off from the third. They write down a set of rules in a precise language about this branching. The rule-set is called C-minimal if, in every world that follows those rules, any group of points you can describe using the language turns out to be simple: just a few whole branches (called cones or balls) put together or cut out of each other. This is like a cousin of another idea, o-minimality, where describable groups on a number line are just a few points and intervals. Some number systems with a way of measuring "closeness," called valued fields, are important examples.

Tame Tree-Geometry Theory

C-minimality is an idea from model theory, the branch of logic that studies what can be expressed about mathematical structures using formal languages. The starting point is a ternary C-relation, a relation among three points that captures a tree-like kind of betweenness: which two points branch together away from the third. A structure with such a relation has natural pieces called cones or balls. A complete theory in this language is C-minimal if, in every model of it (including larger elementary extensions), each subset of the structure definable in one variable is a simple Boolean combination of cones. This is the tree-like counterpart of o-minimality, where definable subsets of an ordered line are finite unions of points and intervals. Algebraically closed valued fields are a major example. To claim a theory is C-minimal, you must be explicit about the language, the C-relation axioms, the notion of cone or ball, and the one-variable definability condition.

 

A C-minimal theory is a complete first-order theory in a language containing a ternary C-relation, satisfying the C-relation axioms, such that in every model the one-variable definable subsets are Boolean combinations of cones (or balls). The C-relation organizes a tree-like betweenness geometry: in one common notation, C(x; y, z) expresses that y and z branch together away from x, and cones are the resulting branch-sets. Requiring the condition in every model, equivalently in all elementary extensions, makes C-minimality a property of the theory rather than of one structure, just as o-minimality is for ordered structures, to which C-minimality is the analogue. Valued fields, especially algebraically closed valued fields, supply the major examples, with balls in the valuation topology playing the role of cones. The identity depends on explicit choices: the language and complete theory, the C-relation axioms, the models and elementary extensions considered, the definition of cones or balls, and the one-variable definable-set condition. Substituting a neighboring notion, such as minimality of a single model, or an ordered-structure analogue, loses the concept.

Scope of Application

C-minimal theory belongs to model theory and is useful where the analyst can specify the typed model theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the language and complete theory, ternary C-relation and axioms, models and elementary extensions, cone or ball definitions, one-variable definable-set condition and examples such as algebraically closed valued fields are explicit. The scope is broad within that domain but bounded by the need for the language and complete theory, ternary C-relation and axioms, models and elementary extensions, cone or ball definitions, one-variable definable-set condition and examples such as algebraically closed valued fields are explicit.

Clarity

The abstraction clarifies a crowded vocabulary by making the language and complete theory, ternary C-relation and axioms, models and elementary extensions, cone or ball definitions, one-variable definable-set condition and examples such as algebraically closed valued fields are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.

Manages Complexity

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to C-minimal theory. C-minimal theory compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning

1. Identify the carrier. State what the elements, states, objects, or observations are: the typed model theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem.
2. Lock the constitutive rule. Express the language and complete theory, ternary C-relation and axioms, models and elementary extensions, cone or ball definitions, one-variable definable-set condition and examples such as algebraically closed valued fields are explicit independently of one notation or implementation.

Knowledge Transfer

Knowledge transfers strongly among subfields of model theory because they reuse the typed model theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, The ternary relation organizes a tree-like betweenness geometry, and minimality restricts definable subsets to simple cone configurations in every model., and type the carrier, state every parameter and convention in the definition, test that the language and complete theory, ternary C-relation and axioms, models and elementary extensions, cone or ball definitions, one-variable definable-set condition and examples such as algebraically closed valued fields are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions

Current abstraction C-minimal theory Domain-specific

Parents (1) — outgoing typed relations

• C-minimal theory is a kind of Minimalism Prime

  It remains its own entry because its identity is fixed by the language and complete theory, ternary C-relation and axioms, models and elementary extensions, cone or ball definitions, one-variable definable-set condition and examples such as algebraically closed valued fields.

Hierarchy paths (2) — routes to 2 parentless roots

• C-minimal theory → is a kind of Minimalism → presupposes Constraint

Show alternative path (1)

Neighborhood in Abstraction Space

C-minimal theory sits in a crowded region of the domain-specific corpus (24^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Order, Type & Classification Foundations (49 abstractions)

Nearest neighbors

• Quasivariety — 0.90
• Transfer principle — 0.89
• Connected relation — 0.89
• Category theory — 0.89
• Join and meet — 0.89

Computed from structural-signature embeddings · 2026-10-25