Imaginaries (model theory)

Imaginaries (model theory)
Name	Imaginaries (model theory)
Field	Mathematics
Subfield	Model theory
Introduced	1970s
Notable	Shelah, Morley, Pillay

Imaginaries (model theory) are a device in mathematical logic that adjoin new sorts to a first-order structure to represent equivalence classes and definable quotients; the formalism underlies canonical parameters, elimination of imaginaries, and many constructions in stability theory. Imaginaries connect syntactic definability with semantic expansions and are central to work by Shelah, Morley, and Pillay on classification and geometric stability. They provide a bridge between model-theoretic properties and concrete structures studied in algebra and geometry.

Definition and basic examples

An imaginary is an element of an expanded structure obtained by adding a new sort for a definable set of equivalence classes; classical expositions appear in papers by Shelah, Morley, and Robinson and in expositions by Pillay and Chang. For example, for a definable equivalence relation E on a definable set D, one adds a sort D/E representing E-classes; this idea features in accounts by Tarski, Skolem, and Zilber and is routine in model-theoretic treatments influenced by Los and Łoś. Common instances include codes for finite sets, tuples modulo permutation, and Galois orbits as studied in work related to Artin, Galois, and Grothendieck.

Equivalence relations and sorts of imaginaries

Imaginaries are organized by definable equivalence relations: given a formula φ(x,y) and parameters from a model M, the relation Eφ partitions the domain and yields a new sort for M/Eφ. This construction is used in canonical parameter arguments by Morley and in elimination proofs by Robinson and Tarski. Important examples appear in algebraic settings connected to Dedekind, Noether, and Hilbert, and in geometric contexts tied to Klein, Riemann, and Weil where definable quotients encode orbits under definable groups like those studied by Lie and Chevalley.

Elimination of imaginaries and canonical parameters

A theory eliminates imaginaries if every imaginary is definably equivalent to a real tuple; canonical parameters for definable sets are the central witness objects. Elimination results feature prominently in work by Macintyre on p-adic fields, by Robinson on algebraically closed fields related to Grothendieck’s ideas, and by Shelah in classification theory. The property is key in applications linking model theory to Zariski geometry, Hasse principles, and Langlands-type considerations where canonical parameters behave analogously to invariants used by Noether, Artin, and Tate.

Applications in stability and classification theory

Imaginaries play a pivotal role in stability, superstability, and simplicity classifications developed by Shelah, Lascar, and Hrushovski; they are essential in defining types, binding groups, and analysability that appear in work by Morley and Baldwin. Tools built from imaginaries enable the analysis of forking, ranks, and orthogonality used in research of Pillay, Ziegler, and Kim. Applications extend to proofs related to the Lachlan program, Vaught’s conjecture contexts, and to connections between model theory and algebraic geometry explored by Zilber, Hrushovski, and Deligne.

Imaginaries in specific theories (o-minimal, algebraically closed fields, etc.)

In o-minimal theories examined by van den Dries, Wilkie, and Hrushovski, imaginaries for definable orders and cell decompositions are tractable and tie to work by Whitney and Thom. Algebraically closed fields eliminate imaginaries via classical geometric invariants tied to Grothendieck, Weil, and Chevalley, and this underpins applications in diophantine geometry influenced by Faltings and Mazur. p-adic fields studied by Macintyre, Denef, and Haskell require additional sorts (leading to elimination relative to value group and residue field) connecting to work by Hasse and Krasner. Differentially closed fields explored by Kolchin and Sit use imaginaries in model-theoretic differential Galois contexts related to Picard–Vessiot theory and to work by Kolmogorov and Ritt.

Technical tools and constructions (groupoids, hyperimaginaries)

Advanced treatments introduce groupoids, hyperimaginaries, and bounded equivalence relations, developed by Lascar, Kim, and Adler, to handle quotients not given by definable equivalence relations. Hyperimaginaries generalize imaginaries to type-definable relations and are instrumental in Lascar strong types, compact group quotients, and in work by Newelski and Wagner on definable topologies. Groupoid approaches connect to Grothendieck’s descent, topos-theoretic viewpoints influenced by SGA, and to categorical formulations used by Gabriel and Zisman, enabling transfer of model-theoretic imaginaries to contexts studied by MacLane, Eilenberg, and Mitchell.

Category:Model theory