Gwyneth University of Leeds Model Theory of Finite and Pseudonite Structures Workshop
Outline 1 Introduction 2 3 4 5
Introduction An asymptotic class is a class of nite structures such that the sizes of the denable sets are uniformly controlled (as a proportion of the size of the ambient structure). I'll be talking about one of several variations of the notion of a 1-dimensional asymptotic class - that of an embedded asymptotic class. This is a work in progress, joint with Dugald Macpherson.
The motivating theorem Theorem (Chatzidakis, van den Dries, Macintyre) Let ϕ(x 1,...,x n ; y 1,...,y m ) be a formula in the language of rings. (i) There is a positive constant C and a nite set D of pairs (d,µ) with d {0, 1,...,n} and µ >0 such that for each nite eld q, where q is a prime power, and each a m, if the set q ϕ( n, a) is nonempty, then for some (d,µ) D q ϕ n, q a d µq < Cq d 1 2. (ii) For every (d,µ) D, there is a formula ψ d,µ (y 1,...,y m ) in the language of rings such that for each q, ψ d,µ m consists of q those a m for which the corresponding inequality holds. q
1-dimensional asymptotic classes Denition (MacPherson and Steinhorn) Let be a rst-order language, and a collection of nite -structures. is a 1-dimensional asymptotic class if for every -formula ϕ(x, y) := ϕ(x 1,...,x n ; y 1,...,y m ): (i) There is a positive constant C and a nite set D of pairs (d,µ) with d {0,...,n} and µ >0, such that for every M and a M m, if ϕ(m n, a) is non-empty, then for some (d,µ) D we have ϕ(m n, a) µ M d C M d 1 2. (ii) For every (d,µ) D, there is an -formula ϕ d,µ (y) such that for all M, ϕ d,µ (M m ) consists of those a M m for which the corresponding inequality holds.
Some notation Let be a rst order language, and a collection of nite -structures. (, y) := (M, a) M, a y M. A partition of (, y) is -denable if for each S there exists an -formula ϕ S (y) without parameters such that for each M. ϕ S (M) = b M y M, b S
Multidimensional asymptotic classes Denition (Ascombe, MacPherson, Steinhorn, Wolf) Let be a class of nite -structures, and R any set of functions 0. is an R-multidimensional asymptotic class (R-MAC) if for every formula ϕ(x, y) there is a nite -denable partition of (, y) and an indexed set H := {h S R S } such that ϕ M x ; b hs (M) = o(h S (M)) for M, b S as M. The functions hs are called the measuring functions of ϕ(x; y). When R is understood, we just say that is a MAC.
Some more notation Let be a rst-order language with signature σ, P a predicate not in σ, and + the language with signature σ {P}. If M is an -structure and A is a (nite) substructure of M, we may view (M, A) as an + -structure (i.e. with P M = A). If ϕ(x 1,...,x n, y 1,...,y m ) is an + formula and a M m, we denote by ϕ M (A n, a) the set b A n : (M, A) ϕ b, a = b : (M, A) P b ϕ b, a.
Denable partitions, renewed If M is an -structure and 0 is a class of nite substructures A of M, let := {(M, A) : A 0 } be the corresponding class of + -structures. As above, for any tuple y we dene (, y) := {(M, A, a) : (M, A), a M y }. We say that a partition of (, y) is -denable if for each S there is an + -formula ϕ S (y) (with no parameters) such that ϕ S (M y ) = {b M y : (M, A, b) S}, for each (M, A).
Embedded MACs Denition Let M be an -structure, and 0 a collection of nite substructures A of M. View each (M, A) as an + -structure, and let = {(M, A) : A 0 }. Let R be any set of functions 0. is an embedded R-MAC if for every + -formula ϕ(x; y) there is a nite -denable partition of (, y) and an indexed set H := {h S R : S } such that ϕ M A x ; b hs (A) = o(h S (A)) for M, A, b S as A.
Remark Let and 0 be as in the previous denition. If is an embedded R-MAC, then 0 is at least a weak R-MAC, meaning that the partition exists but may not be -denable. More assumptions are needed to guarantee that 0 is, in fact, an R-MAC.
Full embeddedness Let N be a -denable -substructure of M, an + -substructure ( + ). N is stably embedded in M if for every + -formula ϕ(x, y) there are nitely many + -formulae ψ 1 (x, z),...,ψ k (x, z) such that for any a M y there are i and b N z for which M x N(ϕ(x, a) ψ i (x, b)). N is canonically embedded in M if for every + -formula ϕ(x) there is an -formula ψ(x) such that for all a N x, M ϕ(a) N ψ(a). N is fully embedded in M if it is both stably and canonically embedded in M.
Transfer lemma Lemma Let, P, and + be as in the denition of an embedded R-MAC. Given an R-MAC 0 consisting of nite substructures of some -structure M, consider the class of + -structures := {(M, A) : A 0 }. If in every ultraproduct (M, P(M )) of, P(M ) is fully embedded in M, then is an embedded R-MAC.
Sketch of proof Fix ϕ(x, y) +. Find nitely many + -formulae ψ 1 (x, z),...,ψ k (x, z) such that for each P(M) 0, (M, P(M)) y z P i x P(ϕ(x, y) ψ i (x, z)). For each i, nd nitely many -formulae ρ i,j (x, z) such that for each P(M) 0, (M, P(M)) j x, z P(ψ i (x, z) ρ P (x, z)). i,j
Sketch of proof, continued Using coding, obtain an -formula η(x; z, w) such that for all P(M) 0 and a M y there are bc P(M) z + w such that ϕ M (P(M), a) = η(p(m), b, c). The above allows us to transfer the partition for 0 to one for. Writing down the dening formulae given those for the original partition is not dicult.
An example The class = {( p, p n) : n } is an embedded 1-dimensional asymptotic class. The class 0 = { p n : n } is a 1-dimensional embedded asymptotic class. Let = rings, + = {P}, and = + {σ} (where σ is a unary function). Consider the following expansion of : = p, p n, x x pn : n. By work of Hrushovski, any ultraproduct (M, P(M ),σ) of is a model of ACFA p. By work of Chatzidakis and Hrushovski, P(M ) is fully embedded in M. This is sucient to apply the lemma.
S. Anscombe, H.D. Macpherson, C. Steinhorn, and D. Wolf, Multidimensional asymptotic classes and generalised measurable structures. In preparation (2016). Z. Chatzidakis, L. van den Dries, and A. Macintyre, Denable sets over nite elds, Journal für die Reine und Angewandte Mathematik, 427, (1992) 107-135. R. Elwes, Asymptotic classes of nite structures, Journal of Symbolic Logic, 72(2), (2007) 418-438. H.D. Macpherson and C. Steinhorn, One-dimensional asymptotic classes of nite structures, Transactions of the American Mathematical Society, 360(1), (2008) 411-448.