Imaginaries in Model Theory Martin Hils Introduction Imaginaries in Model Theory Imaginary Galois theory Algebraic closure in Meq Martin Hils Elimination of imaginaries Geometric Équipe de Logique Mathématique, Université Paris 7 stability Trichotomy Conjecture The group Philosophy and Model Theory Conference configuration Hyperimagi- Université Paris Ouest & École normale supérieure naries June 2-5, 2010, Paris Utility in the unstable case Positive Logic Conclusion What we have learnt Left outs Outline Imaginaries in Model 1 Introduction Theory Martin Hils 2 Imaginary Galois theory Algebraic closure in Meq Introduction Elimination of imaginaries Imaginary Galois theory Algebraic 3 Geometric stability closure in Meq Trichotomy Conjecture Elimination of imaginaries The group configuration Geometric stability Trichotomy 4 Hyperimaginaries Conjecture The group Utility in the unstable case configuration Hyperimagi- Positive Logic naries Utility in the 5 Conclusion unstable case Positive Logic What we have learnt Conclusion What we Left outs have learnt Left outs Context Imaginaries in Model Theory Martin Hils Introduction L is some countable first order language Imaginary Galois theory (possibly many-sorted); Algebraic closure in Meq T a complete L-theory; Elimination of imaginaries U j= T is very saturated and homogeneous; Geometric stability all models M we consider (and all parameter sets A) are Trichotomy Conjecture The group small, with M 4 U. configuration Hyperimagi- naries Utility in the unstable case Positive Logic Conclusion What we have learnt Left outs Imaginary Elements Imaginaries in Model Theory Martin Hils Recall: Introduction An equivalence relation E on a set D is a binary relation Imaginary which is reflexive, symmetric and transitive; Galois theory Algebraic closure in D is partitioned into the equivalence classes modulo E, Meq 0 0 Elimination of i.e. sets of the form d=E := fd 2 D j dEd g. imaginaries Geometric stability Definition Trichotomy Conjecture The group An imaginary element in U is an equivalence class d=E, where configuration E is a definable equivalence relation on a definable set D ⊆ Un Hyperimagi- naries and d 2 D(U). Utility in the unstable case Positive Logic Conclusion What we have learnt Left outs Examples of Imaginaries I Imaginaries in Model Theory Example (Unordered tuples) Martin Hils Introduction In any theory, the formula Imaginary Galois theory (x = x0 ^ y = y 0) _ (x = y 0 ^ y = x0) Algebraic closure in Meq Elimination of 0 0 imaginaries defines an equiv. relation (x; y)E2(x ; y ) on pairs, with Geometric stability 0 0 0 0 Trichotomy (a; b)E2(a ; b ) , fa; bg = fa ; b g: Conjecture The group configuration Thus, fa; bg may be thought of as an imaginary element. Hyperimagi- naries Similarly, for any n 2 , the set fa1;:::; ang may be Utility in the N unstable case Positive Logic thought of as an imaginary. Conclusion What we have learnt Left outs Examples of Imaginaries II Imaginaries in Model Theory Martin Hils k A group (G; ·) is a definable group in U if G ⊆def U and 3 3k Introduction Γ = f(f ; g; h) 2 G j f · g = hg ⊆def U for some k 2 N. Imaginary Galois theory Algebraic Example (Cosets) closure in Meq Elimination of Let (G; ·) be definable group in U and H a definable subgroup imaginaries Geometric of G. Then any coset stability Trichotomy Conjecture g · H = fg · h j h 2 Hg The group configuration Hyperimagi- 0 0 naries is an imaginary (w.r.t. gEg , 9h 2 H g · h = g ). Utility in the unstable case Positive Logic Conclusion What we have learnt Left outs Examples of Imaginaries III Imaginaries in Model Theory Martin Hils Example (Vectors in Affine Space) Introduction Consider the affine space associated to the Q-vector Imaginary n n Galois theory space Q , i.e. the structure M = hQ ; αi, where Algebraic closure in Meq Elimination of α(a; b; c) := a + (c − b): imaginaries Geometric stability The vector bc~ is an imaginary (b; c)=E in M, for Trichotomy Conjecture The group 0 0 0 0 configuration (b; c)E(b ; c ):, α(b; b; c) = α(b; b ; c ): Hyperimagi- naries Utility in the unstable case Positive Logic Conclusion What we have learnt Left outs Utility of Imaginaries Imaginaries in Model Theory Martin Hils Taking into account imaginary elements has several advantages: Introduction Imaginary Galois theory may talk about quotient objects Algebraic closure in Meq (e.g. G=H, where H ≤ G are definable groups) Elimination of imaginaries ) category of def. objects is closed under quotients; Geometric stability Trichotomy right framework for interpretations; Conjecture The group configuration existence of codes for definable sets Hyperimagi- (will be made precise later). naries Utility in the unstable case Positive Logic Conclusion What we have learnt Left outs Adding Imaginaries: Shelah’s Meq-Construction Imaginaries in Model Theory Martin Hils There is a canonical way of adding all imaginaries to M, due to Introduction Shelah, by expanding Imaginary Galois theory Algebraic eq closure in L to a many-sorted language L , Meq Elimination of eq eq imaginaries T to a (complete) L -theory T and Geometric eq eq stability M j= T to M j= T such that Trichotomy eq Conjecture M 7! M is an equivalence of categories between The group eq configuration hMod(T ); 4i and hMod(T ); 4i. Hyperimagi- naries Utility in the unstable case Positive Logic Conclusion What we have learnt Left outs Shelah’s Meq-Construction (continued) Imaginaries in Model Theory For any ;-definable equivalence relation E on Mn we add Martin Hils a new imaginary sort SE Introduction (the intitial sort of M is called the real sort Sreal ), Imaginary n Galois theory a new function symbol πE : Sreal ! SE Algebraic eq closure in ) obtain L ; Meq Elimination of imaginaries axioms stating that πE is surjective, with Geometric stability 0 0 Trichotomy π (a) = π (a ) , aEa Conjecture E E The group configuration eq Hyperimagi- ) obtain T ; naries Utility in the unstable case expand M j= T , interpreting πE and SE accordingly eq Positive Logic eq n M M M ) obtain M = hM; M =E;::: ; R ; f ; : : : ; πE ;:::i. Conclusion What we have learnt Left outs Definable and algebraic closure Imaginaries in Model Theory Definition Martin Hils Let B ⊆ U be a set of parameters and a 2 U. Introduction a is definable over B if fag is a B-definable set; Imaginary Galois theory a is algebraic over B if there is a finite B-definable set Algebraic closure in Meq containing a. Elimination of imaginaries The definable closure of B is given by Geometric stability Trichotomy Conjecture dcl(B) = fa 2 U j a definable over Bg: The group configuration Hyperimagi- Similarly define acl(B), the algebraic closure of B. naries Utility in the unstable case eq Positive Logic These definitions make sense in U ; eq eq eq Conclusion may write dcl or acl to stress that we work in U . What we have learnt Left outs Galois Characterisation of Algebraic Elements Imaginaries in Model Theory Martin Hils Introduction Imaginary Fact Galois theory Algebraic Let AutB (U) = fσ 2 Aut(U) j σ(b) = b 8 b 2 Bg. closure in Meq Elimination of 1 a 2 dcl(B) if and only if σ(a) = a for all σ 2 AutB (U) imaginaries Geometric 2 a 2 acl(B) if and only if there is a finite set A0 containing stability Trichotomy a which is fixed set-wise by every σ 2 AutB (U). Conjecture The group configuration Hyperimagi- naries Utility in the unstable case Positive Logic Conclusion What we have learnt Left outs Existence of codes for definable sets in U eq Imaginaries in Model Theory Fact Martin Hils For any definable set D ⊆ U n there exists c 2 U eq (unique up Introduction to interdefinability) such that σ 2 Aut(U) fixes D setwise if and Imaginary Galois theory only if it fixes c. Algebraic closure in Meq Elimination of imaginaries Proof. Geometric stability Suppose D is defined by '(x; d). Define the equivalence Trichotomy 0 Conjecture relation E(z; z ) as The group configuration 0 Hyperimagi- 8x('(x; z) , '(x; z )): naries Utility in the unstable case Positive Logic Then c := d=E serves as a code for D. Conclusion What we have learnt Left outs The Galois Group Imaginaries eq in Model Any σ 2 AutB (U) fixes acl (B) setwise. Theory Define the Galois group of B as Martin Hils Gal(B) := fσ acleq(B) j σ 2 AutB (U)g: Introduction Imaginary Galois theory Algebraic Example closure in Meq Elimination of imaginaries Let b1 6= b2 be in an infinite set without structure, Geometric stability b := (b1; b2)=E2 (think of b as fb1; b2g) and B = fbg. Trichotomy eq Conjecture Then bi 2 acl (B) and Gal(B) = fid; σg ' Z=2, where σ The group configuration permutes b1 and b2. Hyperimagi- naries Let M j= ACF = T and K ⊆ M a subfield. Then Utility in the alg unstable case Gal(K) = Gal(K =K), where Positive Logic K alg = (field theoretic) algebraic closure of K, Conclusion alg What we Gal(K =K) = (field theoretic) Galois group of K. have learnt Left outs Galois Correspondence in T eq Imaginaries in Model Gal(B) is a profinite group: a clopen subgroup is given by Theory Martin Hils fσ j σ(ai ) = ai 8ig Introduction eq Imaginary for some finite subset fa1;:::; ang of acl (B). Galois theory Algebraic closure in Meq Theorem (Poizat) Elimination of imaginaries There is a 1:1 correspondence between Geometric stability Trichotomy closed subgroups of Gal(B) and Conjecture The group eq eq configuration dcl -closed sets A with B ⊆ A ⊆ acl (B).