COHEIR IN AVERAGEABLE CLASSES

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1 OHEIR IN AVERAGEABLE LASSES WILL BONEY This document serves as a supplement to Boney [Bon, Section 3], in particular to provide the details to Proposition 3.6. Familiarity with that paper and Boney and Grossberg [BG] are assumed and the reader interested in motivation should consult one of those as well. Definition 1. K is averageable iff there is a collection of first order formulas F and Γ such that for all φ j k pj Γ, φ j k F; K N iff for all m and φ(x) F, φ[m] N φ[m] given { i K : i I} and an ultrafilter U on I, Π Γ i /U K and this Γ-ultraproduct satisfies Loś Theorem for the formulas in F. There are two types of coheir to consider. The first is Galois coheir g (or maybe s-coheir 1 ). In this case, we consider Galois types over finite domains. When Galois types are syntactic, these are complete syntactic types over a finite set. The second is t-coheir t, which is more like the first order version. This formulation was not available in [BG] since there was no logic to work with. For averageable classes, F K gives us an appropriate notion of formula to work with. Definition 2. (1) Given A, B,, we say A g B iff for all finite a A, b B, c, gtp(a/bc) is realized in. (2) K has the weak Galois order property iff there are finite tuples a i, b i : i < ω and c and types p q gs(c) such that, for all i, j < ω, j < i = a i b j q j i = a i b j p (3) Given A, B,, we say A t B iff Date: September 3, 2016 Part of this material is based upon work done while the author was supported by the National Science Foundation under Grant No. DS The s and t come from Shelah [Sh:c]; s means the set of parameters is finite and the t means the type is finite. 1

2 2 WILL BONEY for all finite a A, b B, c and φ(x, y, z) F K, if φ(a, b, c), then there is c such that φ(c, b, c). (4) K has the weak order property iff there are finite tuples a i, b i : i < ω and a formula φ(x, y, c) F K with c such that, for all i, j < ω, j < i φ(a i, b j, c) Suppose that averageable K has amalgamation and does not have the weak order property. Note that we have begun talking about Galois types over sets (rather than models, as standard) even though we only have amalgamation over models. This adds some additional dificulties, but we are careful to avoid them here. The adjective weak in describing the order property means that we only require ω length orders, rather than all ordinal lengths as in Shelah [Sh394]. We first work through the properties of g. Proposition 3 ( [BG].4.3g). If is given such that gtp(a/; ) = gtp(a /; ) and a g b and b g a then gtp(ab/; ) = gtp(a b/; ). Note that a, a, b might be finite tuples, not just singletons. Indeed, by the finite character of types and nonforking, this immediately follows for arbitrary sets. Proof: Deny. By < ω-type shortness, there is finite c such that gtp(ab/c; ) = gtp(a b/c; ). Set p = gtp(ab/c; ) and q = gtp(a b/c; ). We build a i, b i : i < ω such that (1) a i b p; (2) j < i, a i b j q; (3) ab i q; and (4) j i, a i b j p. Note that (2) and (4) give the weak order property. onstruction: Suppose we have constructed {a j, b j : j < i} so far. Since a g b, gtp(a/cb{b j : j < i}; ) is realized by some a i. To witness this, there are N and f : cb{bj :j<i} N such that f(a) = a i. This gives gtp(ab/c; ) = gtp(a i b/c; ) gtp(ab j /c; ) = gtp(a i b j /c; ) j < i Similarly, b g a, so gtp(b/ca {a j : j i}; ) is realized by some b i. This gives gtp(a b i /c; ) = gtp(a b/c; ) gtp(a j b i /c; ) = gtp(a j b/c; ) j < i

3 OHEIR IN AVERAGEABLE LASSES 3 Proposition 4 ( [BG].8.2g). g satisfies Extension over models. Proof: Suppose A g N and B N. Because of this nonforking, there is an index set I and an ultrafiler U on I such that, if h : Π Γ /U is the ultrapower map, then h(gtp(a/; N)) is realized in Π Γ /U. all this realization A. Let h + be an isomorphism with range Π Γ N/U that contains h and set Γ,U = (h + ) 1 (Π Γ /U) and N Γ,U N Γ,U Γ,U g similarly. Now we claim B. Let a Γ,U, b B, c. Set p = gtp(a/bc; N) and [f] U = (h + ) 1 (a). Then a p f U h(p) {i I : f(i) p} U 2 Take i 0 from this U-large set. Since [f] U Π Γ /U, we have f(i) and f(i) p. Thus, by onotonicity, (h + ) 1 (A ) g b. Proposition 5 ( [BG].4.1g). (1) If g has Existence and Extension, then it has Symmetry. (2) If g has Symmetry, then it has Uniqueness. Proof: We want to show that A g N B iff B g N A; it is enough to show this for finite a and b. Suppose a g N b. Then, by Existence and Extension, there is some b N such that gtp(b /; N ) = gtp(b/; N) and b g N a. Thus, we have gtp(ab /; N ) = gtp(ab/; N) by 3. By Invariance, b g N a. Now we want to show Uniqueness. Let gtp(a/; N) = gtp(a /; N ), A g N B, and A g N B. By Symmetry, we have B g N A. By 3, we have gtp(ab/; N) = gtp(a B/; N ). Theorem 6 ( [BG].5.1g). If K is an averageable class with amalgamation that doesn t have the weak Galois order property and every model is ℵ 0 -Galois saturated 3, then g is an independence relation in the sense of [BG]. 3 This is equivalent to Existence holding

4 4 WILL BONEY Proof: Existence is provided by the hyptothesis and the rest of the properties follow by the above. Note that the base of g is necessarily a model in Existence, Extension, Symmetry, and Uniqueness. Now we work through the properties of t. Note that the following requires that F is closed under conjunctions and negations. The first is easy to acheive (the F associated to an averageable class can be so closed with no loss), while the second is much harder. However, it holds of our two main cases: Γ-closed and Γ-nice data. Proposition 7 ( [BG].4.3t). Suppose that F is closed under conjunctions and negations. If is given such that tp F (a/; ) = tp F (a /; ) and a t b and b t a then tp F (ab/; ) = tp F (a b/; ). Note that a, a, b might be finite tuples, not just singletons. Indeed, by the finite character of types and nonforking, this immediately follows for arbitrary sets. Proof: Deny. There is c and φ(x, y, c) such that φ(a, b, c) and φ(a, b, c). We build a i, b i : i < ω such that (1) φ(a i, b, c); (2) j < i, φ(a i, b j, c); (3) φ(a, b i, c); and (4) j i, φ(a i, b j, c). Note that (2) and (4) give the weak order property. onstruction: Suppose we have constructed {a j, b j : j < i} so far. Since a t b, φ(x, b, c) j<i φ(x, b j, c) is realized by some a i. In particular, this means φ(a i, b, c) j<i φ(a i, b j, c) Similarly, b g a, so is realized by some b i. This gives φ(a, x, c) j i φ(a j, x, c) φ(a, b i, c) j i φ(a j, b i, c)

5 OHEIR IN AVERAGEABLE LASSES 5 Note that and are interchangeable when the parameters lie in their intersection, which includes b. Proposition 8 ( [BG].8.2t). t satisfies Extension over models. Proof: Suppose A t N and B N. Because of this nonforking, there is an index set I and an ultrafiler U on I such that, if h : Π Γ /U is the ultrapower map, then h(tp(a/; N)) is realized in Π Γ /U. all this realization A. Let h + be an isomorphism with range Π Γ N/U that contains h and set Γ,U = (h + ) 1 (Π Γ /U) and N Γ,U N Γ,U Γ,U g similarly. Now we claim B. Let a Γ,U, b B, c and φ(x, y, z) F K. Set [f] U = (h + ) 1 (a). Then N Γ,U φ(a, b, c) Π Γ N/U φ([f] U, [i b] U, [i c] U ) {i I : N φ(f(i), b, c)} U Take i 0 from this U-large set. Since [f] U Π Γ /U, we have f(i) and satisfies φ(x, b, c). Thus, by onotonicity, (h + ) 1 (A ) g b. Proposition 9 ( [BG].4.1t). (1) If t has Existence and Extension, then it has Symmetry. (2) If t has Symmetry, then it has Uniqueness. Proof: We want to show that A t N B iff B t N A; it is enough to show this for finite a and b. Suppose a t N b. Then, by Existence and Extension, there is some b N such that tp F (b /; N ) = tp F (b/; N) and b t N a. Thus, we have tp F (ab /; N ) = tp F (ab/; N) by 7. By Invariance, b t N a. Now we want to show Uniqueness. Let tp F (A/; N) = tp F (A /; N ), A t N B, and A t N B. By Symmetry, we have B t N A. By 7, we have tp F (AB/; N) = tp F (A B/; N ). Theorem 10 ( [BG].5.1t). If K is an averageable class with amalgamation that doesn t have the weak order property and F is closed under negation and conjunction, then t is an independence relation in the sense of [BG].

6 6 WILL BONEY A crucial point in the above is that we have removed the assumption of Existence (over models). Proof: Existence holds because strong substructre is F-elementary substructure. Then the rest follows from the above propositions. References [Bon] Will Boney, The Γ-Ultraproduct and Averageable lasses, Submitted, org/abs/ [BG] Will Boney and Rami Grossberg, Forking in Short and Tame Abstract Elementary lasses, Submitted. [Sh:c] Saharon Shelah, lassification theory and the number of nonisomorphic models, 2nd ed., vol. 92, North-Holland Publishing o., Amsterdam, xxxiv+705 pp, [Sh394] Saharon Shelah, ategoricity for abstract classes with amalgamation, Annals of Pure and Applied Logic 98 (1990), address: wboney@math.harvard.edu athematics Department, Harvard University, ambridge, A