C-minimal theory¶
A model-theoretic theory whose definable one-variable sets are finite Boolean combinations of cones determined by a ternary C-relation.
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.
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¶
- 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) — more general patterns this builds on
-
C-minimal theory is a kind of Minimalism Prime
The proposed strict upward parent is
prime:minimalism.
Hierarchy paths (2) — routes to 2 parentless roots
- C-minimal theory → Minimalism → Constraint
- C-minimal theory → Minimalism → Abstraction
Neighborhood in Abstraction Space¶
C-minimal theory sits in a moderately populated region (42nd percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Order, Lattices & Set Relations (36 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 · 2026-09-08