12.3 Elimination of imaginaries

Elimination of imaginaries is a powerful technique in model theory that simplifies complex structures. It involves finding explicit definitions for implicitly defined elements, making theories more manageable and revealing hidden properties. This process enhances our understanding of models and their relationships.

By eliminating imaginaries, we can improve a theory's structural properties, simplify the study of definable sets, and uncover geometric or algebraic insights. This technique has wide-ranging applications, from algebraic geometry to group theory, showcasing its importance in mathematical analysis.

Imaginaries in theory

Concept and role of imaginaries

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• Imaginaries represent elements not explicitly defined but implicitly assumed to exist within a theory's framework
• Arise from existential quantifiers in formulas and enhance the theory's expressiveness and completeness
• Function as "ideal" elements extending the original structure, allowing for more general statements and proofs
• Affect decidability and quantifier elimination properties of a theory
• Correspond to definable sets or quotients, representing equivalence classes of definable relations
• Essential for analyzing model-theoretic properties (stability and simplicity)
• Provide insights into geometric and algebraic aspects of a theory's models

Examples and applications

• In algebraic geometry, imaginaries represent points at infinity on projective spaces
• For the theory of algebraically closed fields, imaginaries include roots of polynomials not explicitly defined in the base field
• In the theory of dense linear orders, imaginaries can represent cuts or gaps in the order
• Imaginaries in group theory may represent cosets or quotient groups
• For theories of vector spaces, imaginaries can represent subspaces or linear transformations

Eliminating imaginaries

Process of elimination

• Find explicit definitions for all implicitly defined elements in the theory
• Identify sources of imaginaries (existential quantifiers or definable equivalence relations)
• Construct new sorts or expand language to accommodate explicit representations
• Introduce new function symbols capturing information contained in imaginaries
• Prove every imaginary element is interdefinable with a tuple of elements from the original structure
• Achieve definable bijection between every definable set and a set defined without parameters
• Complexity varies from straightforward to highly intricate procedures depending on the theory

Techniques and strategies

• Analyze definable equivalence relations and construct canonical representatives
• Use compactness arguments to show existence of uniform elimination procedures
• Employ automorphism arguments to demonstrate uniqueness of imaginary representations
• Construct definable Skolem functions to witness existential formulas
• Develop coding schemes to represent complex imaginaries using simpler structures
• Utilize algebraic techniques (Galois theory) for theories with algebraic structures
• Apply model-theoretic tools (saturation, homogeneity) to analyze elimination possibilities

Proving eliminability

Criteria for elimination

• Every definable function can be represented by a definable function to a product of sorts in the original language
• For any formula $\phi(x,y)$ defining a function, a formula $\psi(x,z)$ in the original language uniquely determines the function's value
• Uniform elimination across all models of the theory (uniform elimination of imaginaries)
• Weak elimination up to finite sets (stepping stone to full elimination)
• Every 0-definable equivalence relation has a 0-definable set of representatives
• Theory has enough definable Skolem functions to witness all existential formulas

Proof techniques

• Employ compactness arguments to extend local elimination results to global ones
• Use automorphism arguments to show uniqueness of imaginary representations
• Analyze definable closure properties to establish eliminability
• Construct explicit definitions for imaginaries using the theory's axioms
• Apply induction on formula complexity to prove elimination for all definable sets
• Utilize saturation arguments to extend elimination results to all models
• Develop algebraic techniques (Galois correspondences) for theories with rich algebraic structure

Consequences of elimination

Structural improvements

• Simplifies study of definable sets and functions within the theory
• Enhances stability or simplicity properties of the theory
• Reveals hidden geometric or algebraic structures within theory's models
• Improves quantifier elimination results, potentially leading to decidability
• May transform non-categorical theory into a categorical one
• Facilitates study of type spaces and definable groups within the theory
• Changes automorphism group of models, affecting homogeneity and saturation

Applications and examples

• In algebraically closed fields, elimination leads to effective procedures for solving systems of polynomial equations
• For theories of real closed fields, elimination simplifies the study of semi-algebraic sets
• In differentially closed fields, elimination enables analysis of differential algebraic groups
• Elimination in theories of valued fields improves understanding of definable valuations and henselian fields
• For o-minimal theories, elimination enhances cell decomposition theorems and definable choice functions