O-asymptotic clases of finite structures Daro Alejandro Garca - - PowerPoint PPT Presentation
O-asymptotic clases of finite structures O-asymptotic clases of finite structures Daro Alejandro Garca Universidad de los Andes XVI Simposio LatinoAmericano de Lgica Matemtica Julio 28 - Agosto 1 / 2014 Buenos Aires - Argentina 1 / 1
O-asymptotic clases of finite structures
Darío Alejandro García Universidad de los Andes XVI Simposio LatinoAmericano de Lógica Matemática Julio 28 - Agosto 1 / 2014 Buenos Aires - Argentina
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O-asymptotic clases of finite structures
ACF Th(Q, <) (R, +, ·, <) Random Graph Pseudofinite fields Peano Arithmetic
Simple NIP Stable
Strongly minimal
Rosy NTP2
O-minimal A more detailed map at www. forkinganddividing. com (due to Gabriel Conant) 2 / 1
O-asymptotic clases of finite structures 3 / 1
O-asymptotic clases of finite structures 1-dimensional asymptotic classes of finite structures
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O-asymptotic clases of finite structures 1-dimensional asymptotic classes of finite structures Definition
Definition (1-dimensional asymptotic class) Let L be a first order language, and C be a collection of finite L-structures. Then C is a 1-dimensional asymptotic class if the following hold formula ϕ(x, y), where y = (y1, · · · , ym):
1 There is a positive constant C and a finite set E ⊆ R>0 such
that for every M ∈ C and a ∈ Mm, either |ϕ(M, a)| ≤ C, or for some µ ∈ E, ||ϕ(M, a)| − µ|M|| ≤ C|M|1/2 (∗)
2 For every µ ∈ E, there is an L-formula ϕµ(y) such that for all
M ∈ C, ϕµ(Mm) is precisely the set of a ∈ Mm where (*) holds.
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O-asymptotic clases of finite structures 1-dimensional asymptotic classes of finite structures Definition
Definition (1-dimensional asymptotic class) Let L be a first order language, and C be a collection of finite L-structures. Then C is a 1-dimensional asymptotic class if the following hold formula ϕ(x, y), where y = (y1, · · · , ym):
1 There is a positive constant C and a finite set E ⊆ R>0 such
that for every M ∈ C and a ∈ Mm, either |ϕ(M, a)| ≤ C, or for some µ ∈ E, ||ϕ(M, a)| − µ|M|| ≤ C|M|1/2 (∗)
2 For every µ ∈ E, there is an L-formula ϕµ(y) such that for all
M ∈ C, ϕµ(Mm) is precisely the set of a ∈ Mm where (*) holds.
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O-asymptotic clases of finite structures 1-dimensional asymptotic classes of finite structures Main Examples
(1) The class of finite fields: This is essentially the remarkable theorem of Chatzidakis-Van den Dries - Macyintire about definable sets in finite and pseudofinite fields. (2) The class of Paley graphs:The proof of this uses the following result due to Bollobás-Thomason Theorem Let U and W be disjoint sets of vertices of the Paley graph Pq with |U ∪ W | = m, and let v(U, W ) be the number of vertices not in U ∪ W joined to each vertex of U and no vertex of W ; then |v(U, W ) − 2−mq| ≤ 1 2(m − 2 + 2−m+1)q1/2 + m 2
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O-asymptotic clases of finite structures 1-dimensional asymptotic classes of finite structures Main Examples
(1) The class of finite fields: This is essentially the remarkable theorem of Chatzidakis-Van den Dries - Macyintire about definable sets in finite and pseudofinite fields. (2) The class of Paley graphs:The proof of this uses the following result due to Bollobás-Thomason Theorem Let U and W be disjoint sets of vertices of the Paley graph Pq with |U ∪ W | = m, and let v(U, W ) be the number of vertices not in U ∪ W joined to each vertex of U and no vertex of W ; then |v(U, W ) − 2−mq| ≤ 1 2(m − 2 + 2−m+1)q1/2 + m 2
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O-asymptotic clases of finite structures 1-dimensional asymptotic classes of finite structures Main Examples
(3) The class of finite cyclic groups: This is proved by using the Szmielew’s Theorem about definable sets in cyclic groups to get a quantifier elimination result, and then calculating the possible measures of sets in
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O-asymptotic clases of finite structures 1-dimensional asymptotic classes of finite structures Main Examples
(3) The class of finite cyclic groups: This is proved by using the Szmielew’s Theorem about definable sets in cyclic groups to get a quantifier elimination result, and then calculating the possible measures of sets in
(4) Non-example: The class of finite linear orders. The formula x < a can pick up an initial segment of arbitrary size as a varies.
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O-asymptotic clases of finite structures 1-dimensional asymptotic classes of finite structures Ultraproducts of asymptotic classes
Theorem (Macpherson-Steinhorn) Let C be a class of finite structures. If every infinite ultraproduct of members of C is strongly minimal, then C is a 1-dimensional asymptotic class. Theorem (Macpherson-Steinhorn) Suppose C is a 1-dimensional asymptotic class, and let M be an infinite ultraproduct of members of C. Then, Th(M) is supersimple
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O-asymptotic clases of finite structures 1-dimensional asymptotic classes of finite structures Ultraproducts of asymptotic classes
Theorem (Macpherson-Steinhorn) Let C be a class of finite structures. If every infinite ultraproduct of members of C is strongly minimal, then C is a 1-dimensional asymptotic class. Theorem (Macpherson-Steinhorn) Suppose C is a 1-dimensional asymptotic class, and let M be an infinite ultraproduct of members of C. Then, Th(M) is supersimple
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O-asymptotic clases of finite structures O-asymptotic classes of finite structures
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O-asymptotic clases of finite structures O-asymptotic classes of finite structures Definition
Definition (O-asymptotic classes of finite structures) Let C be a class of finite linearly ordered structures in a language L containing <. We say C is a O-asymptotic class if for every formula ϕ(x; y1, . . . , ym) there is a constant C > 0 and k ≥ 1 and a finite set E ⊆ ([0, 1])k such that:
c0 = min M < c1 < . . . < ck = max M and a tuple µ ∈ E such that: (*) For every i = 1, 2, . . . , k, either µi = 0 and |ϕ(M, a) ∩ (ci−1, ci)| ≤ C
µi > 0 and for every (u, v) ⊆ (ci−1, ci), ||ϕ(M, a) ∩ (u, v)| − µi|(u, v)|| ≤ C|(u, v)|1/2
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O-asymptotic clases of finite structures O-asymptotic classes of finite structures Definition
Definition (continuation)
M | = ϕµ(a; c1, . . . , ck) implies (*) holds The main difference is that we require that every formula in one variable admits a decomposition into a uniform number of intervals such that on each interval it satisfies a distribution like in 1-dimensional classes, and the set is distributed uniformly.
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O-asymptotic clases of finite structures O-asymptotic classes of finite structures Two examples
Example The class of finite linear orders. The definable sets in one variable are (uniformly) finite union
The measures in this case will be tuples µ = {0, 1}k ⊆ [0, 1]k.
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O-asymptotic clases of finite structures O-asymptotic classes of finite structures Two examples
Example The class of finite linear orders. The definable sets in one variable are (uniformly) finite union
The measures in this case will be tuples µ = {0, 1}k ⊆ [0, 1]k.
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O-asymptotic clases of finite structures O-asymptotic classes of finite structures Two examples
Example The class of finite linear orders. The definable sets in one variable are (uniformly) finite union
The measures in this case will be tuples µ = {0, 1}k ⊆ [0, 1]k.
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O-asymptotic clases of finite structures O-asymptotic classes of finite structures Two examples
Let CZ = {ZN : N < ω} be the class of finite structures where ZN = (Z/(2N + 1)Z, +, <N) where (Z/(2N + 1)Z, +) is the cyclic group with 2N + 1 elements and <N is the order given by −N < −(N − 1) < · · · < −1 < 0 < 1 < · · · N − 1 < N Example The class CZ is an O-asymptotic class
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O-asymptotic clases of finite structures O-asymptotic classes of finite structures Two examples
Let CZ = {ZN : N < ω} be the class of finite structures where ZN = (Z/(2N + 1)Z, +, <N) where (Z/(2N + 1)Z, +) is the cyclic group with 2N + 1 elements and <N is the order given by −N < −(N − 1) < · · · < −1 < 0 < 1 < · · · N − 1 < N Example The class CZ is an O-asymptotic class
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O-asymptotic clases of finite structures O-asymptotic classes of finite structures Ultraproducts of O-asymptotic classes
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O-asymptotic clases of finite structures O-asymptotic classes of finite structures Ultraproducts of O-asymptotic classes
Theorem (G.) Let C be a class of finite linearly ordered structures. If every infinite ultraproduct of members of C is O-minimal, then C is an O-asymptotic class.
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O-asymptotic clases of finite structures O-asymptotic classes of finite structures Ultraproducts of O-asymptotic classes
Theorem (G.) Let C be a class of finite linearly ordered structures. If every infinite ultraproduct of members of C is O-minimal, then C is an O-asymptotic class. Theorem (G.) Suppose C is an O-asymptotic class, and let M be an infinite ultraproduct of members of C. Then,
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O-asymptotic clases of finite structures O-asymptotic classes of finite structures Ultraproducts of O-asymptotic classes
Theorem (G.) Let C be a class of finite linearly ordered structures. If every infinite ultraproduct of members of C is O-minimal, then C is an O-asymptotic class. Theorem (G.) Suppose C is an O-asymptotic class, and let M be an infinite ultraproduct of members of C. Then, Th(M) is superrosy of Uþ-rank 1.
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O-asymptotic clases of finite structures O-asymptotic classes of finite structures Ultraproducts of O-asymptotic classes
Theorem (G.) Let C be a class of finite linearly ordered structures. If every infinite ultraproduct of members of C is O-minimal, then C is an O-asymptotic class. Theorem (G.) Suppose C is an O-asymptotic class, and let M be an infinite ultraproduct of members of C. Then, Th(M) is superrosy of Uþ-rank 1. M is NTP2 (in fact, inp-minimal)
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O-asymptotic clases of finite structures Further results
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O-asymptotic clases of finite structures Further results
Theorem (Macpherson-Steinhorn) Suppose that C is a 1-dimensional asymptotic class of finite L-structures. Then, for every formula ϕ(x, y) ∈ L with |x| = n, |y| = m we have:
1 There is a constant C > 0 and a finite set D of pairs (d, µ)
with d ∈ {0, 1, . . . , n} and µ ∈ R>0 such that for every M ∈ C, a ∈ Mm there is some(d, µ) ∈ D such that ||ϕ(Mn; a) − µ|M|d| ≤ C|M|d−1/2 (∗∗)
2 For every (d, µ) ∈ D, there is an L-formula ϕ(d,µ)(y) such that
M | = ϕ(d,µ)(a) if and only if (∗∗) holds
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O-asymptotic clases of finite structures Further results
Theorem (Macpherson-Steinhorn) Suppose that C is a 1-dimensional asymptotic class of finite L-structures. Then, for every formula ϕ(x, y) ∈ L with |x| = n, |y| = m we have:
1 There is a constant C > 0 and a finite set D of pairs (d, µ)
with d ∈ {0, 1, . . . , n} and µ ∈ R>0 such that for every M ∈ C, a ∈ Mm there is some(d, µ) ∈ D such that ||ϕ(Mn; a) − µ|M|d| ≤ C|M|d−1/2 (∗∗)
2 For every (d, µ) ∈ D, there is an L-formula ϕ(d,µ)(y) such that
M | = ϕ(d,µ)(a) if and only if (∗∗) holds
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O-asymptotic clases of finite structures Further results
Theorem (G.) Suppose C is an O-asymptotic class of finite L-structures. Then for every ϕ(x; y) (|x| = n, |y| = m) the following hold: There is a positive constant C > 0, k = k(ϕ) and a finite set of tuples α ∈ [0, 1]k such that for every M ∈ C and a ∈ Mm there is a cell-decomposition {Z1, . . . , Zk} of Mn into k cells such that: (∗)k : For 1 ≤ i ≤ k, either αi = 0 and |ϕ(Mn, a) ∩ Zi| ≤ C
αi > 0 and ||ϕ(M, a) ∩ Zi| − αi|Zi|| ≤ C|Li|dim Zi−1/2 where Li is a cell of dimension 1 and maximal size contained in Zi.
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O-asymptotic clases of finite structures Further results
1 Multiplication by an irrational factor:
For a fixed α ∈ [0, 1] − Q, consider the class Cα = {Mn = ([0, n], <, f ) : n < ω} where f (x) := α · x in every structure.
2 Is there an O-asymptotic class with an independent
ultraproduct? My guess is it is possible to have a class of finite linearly
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O-asymptotic clases of finite structures Further results
1 Multiplication by an irrational factor:
For a fixed α ∈ [0, 1] − Q, consider the class Cα = {Mn = ([0, n], <, f ) : n < ω} where f (x) := α · x in every structure.
2 Is there an O-asymptotic class with an independent
ultraproduct? My guess is it is possible to have a class of finite linearly
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O-asymptotic clases of finite structures Further results
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O-asymptotic clases of finite structures Further results
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O-asymptotic clases of finite structures References
February 2014, Pages 695–723
Symbolic Logic. Volume 72, Issue 2 (2007), 418-438.
classes of finite structures. Transactions Of The American Mathematical Society. v.360 Number.1 p.411 - 448 ,2008
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O-asymptotic clases of finite structures References
(No. 379). Cambridge University Press. 2011.
London Mathematical Society. Lecture Note Series 248. Cambridge University Press. 1998
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