Elimination of imaginaries

Elimination of imaginaries
Name	Elimination of imaginaries
Field	Model theory
Introduced	1960s
Notable people	André Weil; Abraham Robinson; Alfred Tarski; James Ax; Boris Zilber; Angus Macintyre; Ehud Hrushovski
Related concepts	Model theory; Stability theory; Categoricity; O-minimality

Elimination of imaginaries

Elimination of imaginaries is a property in model theory concerning definable equivalence relations and canonical parameters for definable sets. It refines notions studied by André Weil, Abraham Robinson, Alfred Tarski, James Ax and Boris Zilber and plays a role in interactions among Algebraic geometry, Number theory, Group theory, Set theory and Logic.

Definition and basic examples

A theory T has elimination of imaginaries when every definable equivalence class in models of T has a canonical parameter definable in the home sort; this notion arose in work by André Weil, Abraham Robinson, Alfred Tarski and James Ax and is linked to notions in Algebraic geometry by Grothendieck-style parameter spaces. Classic examples include algebraically closed fields (work of André Weil and later model-theoretic treatments by Ehud Hrushovski and Boris Zilber), real closed fields related to Alfred Tarski’s decidability results, and vector spaces over division rings considered by Abraham Robinson and Angus Macintyre. Counterexamples and partial results appear in research by Ehud Hrushovski, Boris Zilber and other participants in the model-theoretic community around Institute for Advanced Study and universities such as University of Oxford and Hebrew University of Jerusalem.

Historical development and motivations

The motivation traces through André Weil’s use of parameter spaces in Algebraic geometry and Alfred Tarski’s decision method for Real closed fields; Abraham Robinson’s work on nonstandard analysis and James Ax’s research on finite fields influenced formal model-theoretic framing. Developments in the 1960s and 1970s at institutions like the University of California, Berkeley and Princeton University connected elimination of imaginaries to stability theory pioneered by Shelah and later refined in seminars involving Boris Zilber, Ehud Hrushovski and Angus Macintyre. Applications to problems posed by Alexander Grothendieck and connections to results by Jean-Pierre Serre, Pierre Deligne and Alexander Grothendieck clarified the algebraic geometry perspective, while links to work by Saharon Shelah, Wilfrid Hodges and Bruno Poizat shaped the logical formalism.

Formal definitions and equivalent formulations

Formally, T eliminates imaginaries if for every definable set X and definable equivalence relation E there is a definable function f whose fibers are E-classes, a perspective developed in model-theoretic treatments by Saharon Shelah, Bruno Poizat and Wilfrid Hodges. Equivalent formulations involve expansion by sorts for quotient sets, canonical parameter sorts introduced in expositions by Angus Macintyre and Ehud Hrushovski, and the existence of imaginary sorts making the category of definable sets well-powered as in expositions influenced by Alexander Grothendieck and Jean-Pierre Serre. Connections to definable Galois groups in the sense of James Ax and Boris Zilber provide alternative characterizations used in stability-theoretic accounts by Saharon Shelah.

Examples in specific theories

Algebraically closed fields (results tied to André Weil and Ehud Hrushovski) eliminate imaginaries after adding sorts for projective spaces studied by Alexander Grothendieck and Jean-Pierre Serre; real closed fields (Tarski) require named constants or Dedekind-complete expansions studied by Alfred Tarski and Angus Macintyre; differentially closed fields examined by Abraham Robinson and Michael Singer need additional parameter sorts as in work by Ehud Hrushovski and Anand Pillay; separably closed fields and pseudofinite fields considered by James Ax and Ehud Hrushovski often eliminate imaginaries only up to finite covers, a theme in Boris Zilber’s research on fields and groups. Modules and vector spaces over division rings studied by Abraham Robinson and Angus Macintyre often have full elimination of imaginaries with canonical bases appearing in work by Saharon Shelah.

Preservation and transfer results

Preservation under expansions and reducts has been studied by Saharon Shelah, Bruno Poizat and Wilfrid Hodges; adding constants, definable Skolem functions or sorts (as in constructions used by Angus Macintyre and Ehud Hrushovski) can force elimination of imaginaries, while reducts studied by Abraham Robinson and Alfred Tarski may lose the property. Transfer along bi-interpretability and conservative extensions appears in research by Boris Zilber, James Ax and Jean-Pierre Serre; model companions and completions treated by Saharon Shelah and Ehud Hrushovski give criteria for when elimination of imaginaries passes to companions, and group configuration techniques by Ehud Hrushovski and Boris Zilber yield structural transfer results.

Applications in model theory and algebraic geometry

In model theory applications, elimination of imaginaries underpins definable group theory in works of Boris Zilber, Ehud Hrushovski and Anand Pillay, and it facilitates the development of Galois-like correspondences as in James Ax’s and Jean-Pierre Serre’s perspectives. In algebraic geometry, canonical parameters relate to moduli problems treated by Alexander Grothendieck and Jean-Pierre Serre, and interactions with cohomology theories studied by Pierre Deligne and Alexander Grothendieck exploit elimination-type phenomena for definable sheaves. Number-theoretic applications draw on James Ax’s and Ehud Hrushovski’s analyses of pseudofinite fields and Diophantine geometry questions addressed by Alexander Grothendieck and Jean-Pierre Serre.

Connections to stability and o-minimality

Stability theory developed by Saharon Shelah and later work by Boris Zilber and Ehud Hrushovski ties elimination of imaginaries to canonical base theory, internality and binding groups; these themes appear in Saharon Shelah’s classification theory and in Boris Zilber’s categoricity conjectures. In o-minimality studied by Lou van den Dries, Alfred Tarski and Angus Macintyre, elimination of imaginaries often follows from cell decomposition and definable choice results by Lou van den Dries and Christopher Miller, with further refinements by Anand Pillay and Charles Steinhorn.

Category:Model theory