ATOMIC SATURATION OF REDUCED POWERS 6 SAHARONSHELAH 1 0 2 Abstract. Ouraimwastogeneralizesometheoremsaboutthesaturationof ultra-powers to reduced powers. Naturally, we deal via saturation for types n consisting of atomic formulas. We succeed to generalize “the theory of dense a linearismaximalandsoisanypair(T,∆)whichisSOP3(∆consistsofatomic J or conjunction of atomic formulas). However, SOP2 is not enough, so the 9 p=ttheoremcannotbegeneralizedinthiscase. Similarlytheuniquedualof 1 cofinality. ] O L . h t a m [ 1 v 4 2 8 4 0 . 1 0 6 1 : v i X r a Date:September 28,2015. 2010 Mathematics Subject Classification. Primary: 03C20;Secondary: 03C45,03C75. Key words and phrases. model theory, set theory, reduced power, saturation, classification theory,SOP3,SOP2. The author thanks Alice Leonhardt for the beautiful typing. First typed February 26, 2014. Paper1064. 1 2 SAHARONSHELAH Anotated Content §0 Introduction, (labels x,z), pg.3 §1 Axiomatizing [Sh:c, Ch.VI,2.6], (label h), pg. 8 [We phrase and prove a theorem which axiomatize [Sh:c, Ch.VI,2.6]. The theoremtheresaysthatifDisaregularultrafilteronI andforeverymodel M of the theory of dense linear orders, the model MI/D is λ+-saturated, then D is λ+-good and λ-regular.] §2 Applying the axiomatized frame, (label c), pg.12 [The axiomatization in §1 can be phrased as a set of sentences, surpris- ingly moreover Horn ones (first order if θr =ℵ0). Now in this case we can straightforwardlyderive[Sh:a, Ch.VI,2.6]. But because the axiomatization being Horn, we can now deal also with the (λ+,atomic)-saturation of reduced power. We then dealwith infinitary logics and comments on models of Bounded Peano Arithmetic.] §3 Criterion for atomic saturation of reduced powers, (label g), pg.18 [For a complete first order T we characterize when a filter D on I is such that MI/D is (λ,atomically)-saturated for every model M of T.] §4 Counterexample, (label b), pg.23 [We prove that for reduced powers, the parallel of t ≤ p in general fails, similarly the uniqueness of the dual cofinality.] §5 The orders Erp, (label g,r), pg.32 λ [We define and prove the results on morality from the ultra-filter case to the filter case. By this weprovethatsomeSOP pairarenotErp-maximal 2 (unlike the ultra-filter cases, see 5.10). ATOMIC SATURATION OF REDUCED POWERS 3 § 0. Introduction § 0(A). Background, Questions and Answers. We know much on saturation of ultrapowers, but considerably less on reduced powers. Fortransparency,letT denoteafirstordercompletecountabletheorywith elimination of quantifiers and M will denote a model of T. For D a regular filter on λ > ℵ we may ask: when Mλ/D is saturated? For D an ultrafilter, Keisler 0 [Kei65] proves that this holds for every T iff D is λ+-good iff this holds for T = theory of Boolean algebras, such T is called E -maximal. λ By [Sh:a, Ch.VI,2.6] the maximality holds for T = theory of dense linear orders or just any T with the strict orderproperty and by [Sh:500], any T with the SOP 3 is E -maximal. λ What about reduced powers for λ-regular filter D on λ? By [Sh:17], Mλ/D is λ+-saturatedfor everyT iff D is λ+-goodandP(λ)/D is a λ+-saturatedBoolean Algebra. Parallelresultsholdwhenwereplaceλ+-saturatedby(λ+,Σ1+n(Lτ(T)))- saturated. We shall concentrate on (λ+,atomically)-saturatedand the related partial order Erp, see definitions below. λ Concerningultra-powers,lately Malliaris-Shelah[MiSh:998]provesthatT being SOP sufficefor⊳ -maximality(andthatt=p)and,inlaterwork[MiSh:1051],that 2 λ atleastfora relative⊳∗ (see[Sh:500])this is “iff”. Partofthe proofis axiomatized λ by Malliaris-Shelah [MiSh:1070]. Note also that [Sh:1019] deals with saturation but only for ultra-powers by θ- complete ultrafilters for θ a compact cardinal; and also with ω-ultra-limits. Now what do we accomplish here? First, in §1 we axiomatize the proof of [Sh:a, Ch.VI,2.6], i.e. we define when r=(M,∆)isanRSPandforitprovethattherelevantmodelNris(min{pr,tr},∆)- saturated. Second,in§2weprove,ofcourse,that[Sh:a,Ch.VI,2.6]follows,butalso weshowthatthe axiomatizationofRSP isby Hornsentences. Hence wecanapply it to reduced powers. So T is Erp-maximal if T = Th(Q,<) and for T having the λ SOP ; lastly we comment on models of Peano Arithemetic. 3 In §3 we try to sort out when for models of T we get the relevant atomic saturation. Canwegeneralizealsoresults[MiSh:998]toreducedpowers? Themainresultof §4 says that no. We also sort out the parallelof goodness, excellency and morality for filters and atomic saturation for reduced powers. In hopeful continuation, we considerparallelstatementsforinfinite logics(see[Sh:1019]); alsoweconsidernon- maximality. Note that by 2.10 Conclusion 0.1. If (T,∆) has the SOP , then it is Erp-maximal. 3 λ Question 0.2. Do we have: ifD is (λ ,T)-goodandregularthen D is (λ ,T)-good 2 1 when λ <λ (or more). 1 2 4 SAHARONSHELAH § 0(B). Further Questions. Convention 0.3. 1) Let T be a theory with elimination of quantifiers if not said otherwise. Let Mod be the class of models of T. T 2)ThemaincaseisforT isacountablecompletefirstordertheorywithelimination of quantifiers, moreover,with every formula equivalent to an atomic one. So it is natural to ask Conjecture 0.4. (T,∆) is E -maximal iff (T,∆) has the SOP . rp 3 So which T (with elimination of quantifiers) are maximal under ⊳rp? That is, λ when for every regular filter D on λ,Mλ/D is (λ+, atomically)-saturated iff D is λ+-good? Is T maximal? As we have not proved this even for ultrafilters, the feq reasonable hope is that it will be easier to show non-maximality for ⊳rp. Also in λ light of [MiSh:1030] for simple theories we like to prove non-maximality with no large cardinals. We may hope to use just NSOP , but still it would not settle the 2 problem of characterizing the maximal ones as, e.g. SOP ≡ SOP is open; for 2 3 such T; for a pair (T,ϕ(x¯,y¯)) they are different. Note that for first order T, it makes sense to use µ+-saturated models and D is µ+-complete. Also the “T stable” case should be resolved. Conjecture 0.5. Mλ/D is (ℵλ/D,atomically)-saturatedwhen: 0 (a) T a theory as in 0.3 (b) T is stable without the fcp (c) D is a (λ,ℵ )-regular filter on λ. 0 Remark 0.6. Maybegivena1−ϕ-typep⊆{ϕ(x,a¯):a¯∈m(MI/D)}ofcardinality ≤λ in MI/D, we try just to find a dense set of A∈D+ such that in MI/(D+A) the 1−ϕ-type is realized. Then continue; opaque. § 0(C). Preliminaries. Notation 0.7. 1) Let B denote a Boolean algebra, comp(B) its completion, B+ = B\{0 },uf(B) the setofultrafilters onB,fil(B) the setoffilters onB. For a∈B B let aif(true) =aif(1) be a and let aif(false) =aif(0) be 1B−a. 2) For a model M let τ =τ(M) be its vocabulary. M Now about cuts (they are different than gaps, see [MiSh:1069]). Definition 0.8. 1) For a partial order T = (T,≤T), we say (C1,C2) is pre-cut when: (a) C1∪C2 is a subset of T linearly ordered by ≤T (b) if a1 ∈C1,a2 ∈C2 then a1 ≤T a2 (c) for no c∈T do we have a1 ∈C1 →a1 ≤T c and a2 ∈C2 ⇒c≤T a2. 2) Above we say (C ,C ) is a (κ ,κ )-pre-cut when in addition: 1 2 1 2 ATOMIC SATURATION OF REDUCED POWERS 5 (d) C has cofinality κ 1 1 (e) C∗, the inverse of C , has cofinality κ 2 2 2 (f) so κ ,κ are regular or 0 or 1. 1 2 3) We may replace Cℓ by a sequence a¯ℓ, if not said otherwise such that a¯1 is ≤T- increasing and a¯2 is ≤T-decreasing. 4) We say (C ,C ) is a (κ ,κ )-linear-cut of T when it is a (κ ,κ )-pre-cut and 1 2 1 2 1 2 C ∪C is downward closed, so natural for T a tree. 1 2 5) We say (C ,C ) is a weak cut when (b),(c) of part (1) holds. 1 2 6) We may write cut instead of pre-cut. Remark 0.9. IfT isa(modeltheoretic)tree, κ >0 and(C ,C )is a(κ ,κ )-pre- 2 1 2 1 2 cutthen itinducesoneandonlyone(κ ,κ )-linear-cut(C′,C′),i.e. onesatisfying 1 2 1 2 C ⊆C′,C ⊆C′ such that C ∪C is cofinal in C′ ∪C′. 1 1 2 2 1 2 1 2 Definition 0.10. 1) We say M is fully (λ,θ,σ,L)-saturated (may omit the fully); where L⊆L(τ ) and L is a logic; we may write L if L=L(τ ), when: M M • if Γ is a set of <λ formulas from L with parameters from M with <1+σ free variables, and Γ is (<θ)-satisfiable in M, then Γ is realized in M. 2) We say “locally” when using one ϕ = ϕ(x¯,y¯) ∈ L with ℓg(x¯) < 1+σ, i.e. all members of Γ have the form1 ϕ(x¯,¯b). 3) Saying “locally/fully (λ,L)-saturated” the default values (i.e. we may omit) of σ is σ =θ, of (σ,θ) is θ =ℵ ∧σ =ℵ and of L is L (first order logic) and of L is 0 0 L. Omitting L means Lθ,θ, omitting λ means λ=kMk. 4) If ϕ(x¯,y)∈L(τ ) and a¯∈ℓg(y¯)M then ϕ(M,a¯):={¯b∈ℓg(x¯)M :M |=ϕ[¯b,a¯]}. M 5) Let x¯ =hx :s∈ui. [u] s Definition 0.11. AssumewearegivenaBooleanAlgebraBusuallycompleteand a model or a set M and D a filter on comp(B), the completion of B. 1)Let MB be the setofpartialfunctions f fromB+ :=B\{0B} into M suchthat for some maximal antichain ha : i < i(∗)i of B,Dom(f) includes {a : i < i(∗)} i i and is included in2 {a ∈ B+ : (∃i)(a ≤ a )} and f is a function into M and i f↾{a∈Dom(f):a≤a } is constant for each i. i 1A) Naturally for f ,f ∈ MB we say f ,f are D-equivalent, or f = f mod D 1 2 1 2 1 2 when for some b∈D we have a1 ∈Dom(f1)∧a2 ∈Dom(f2)∧a1∩a2∩b>0B ⇒ f (a )=f (a ). 1 1 2 2 2) We define MB/D naturally, as well as TV(ϕ(f ,...,f )) ∈ comp(B) when 0 n−1 ϕ(x0,...,xn−1)∈L(τM) and f0,...,fn−1 ∈MB where TV stands for truth value and MB/D|=ϕ[f /D,...,f /D] iff TV (ϕ(f ,...,f ))∈D. 0 n−1 M 0 n−1 2A) Abusing notation, not only MB1 ⊆ MB2 but MB1/D1 ⊆ MB2/D2 when B1⋖B2,Dℓ ∈fil(Bℓ)forℓ=1,2andD1 =B1∩D2. Also[f1,f2 ∈MB1 ⇒f1 =f2 mod D1 ↔f1 =f2 mod D2]. So for f ∈MB1 we identify f/D1 and f/D2. 1In [Sh:1019] weuseaL⊆Lθ,θ,θ acompact cardinaland ifσ>θ weuseaslightlydifferent versionofthedefinitionoflocalandofthedefaultvaluesofσ wasθ. 2for the Dℓ ∈ uf(Bℓ) ultra-product, without loss of generality B is complete, then without loss of generality f↾{ai : i < i(∗)} is one to one. But in general we allow ai = 0B, those are redundantbutnaturalin0.11(3). 6 SAHARONSHELAH 3) For complete B, we say han : n < ωi represents f ∈ NB when han : n < ωi is a maximal antichain of B (so an = 0B is allowed) and for some f′ ∈ NB which is D-equivalent to f (see 0.11(1A)) we have f′(a )=n. n 4) We say h(an,kn):n<ωi represent f ∈NB when: (a) the k are natural numbers with no repetition n (b) ha :n<ωi is a maximal antichain n (c) f(a )=k . n n 5) If I is a maximal antichain of B and M¯ = hM : a ∈ Ii is a sequence a of τ-models, then we define M¯B be the set of partial functions f from B+ to ∪{M : a ∈ I} such that for some maximal antichain ha : i< i(∗)i of B refining a i I (i.e. (∀iℵ0 then in (T,≤T) any increasingchain of length <θ whichhas an upper bound has a ≤T-lub. Remark 1.3. Wemayconsideradding: SM abeingsuccessor,(butthisisnotHorn), i.e.: (l) if ι=1 we also have SM ={(a,b):B is a ≤T-successor of a such that (α) if a≤b∧a6=b then for some c,S(a,c)∧c≤b (β) if b∈T\{rtT} then for some unique a we have SM(a,b) (γ) S(a,b)⇒a≤b (δ) S(a,b )∧S(a,b )∧b 6=b ⇒¬(b ≤b ) 1 2 1 2 1 2 (ε) in clause (j) we can add SM(s,t). Remark 1.4. Presently, it may be that a ≤T b ≤T a but a 6= b. Not a disaster to forbid but no reason. How does this axiomatize realizations of types? Claim/Definition 1.5. Let ι={1,2},θ is ℵ or just a regular cardinal. 0 1) For any model N and ∆ ⊆ {ϕ= ϕ(x,y¯) ∈Lθ,θ(τT)} closed under conjunctions of <θ, the canonical (θ,ι)−RSP,r=rθ defined below is indeed a θ−RSP. N,∆ 2) r=rθ (if θ =ℵ we may omit it) is defined by: N,∆ 0 (a) ∆r =∆,Nr =N and θr =θ (b) Tr = {hϕε(x,a¯ε) : ε < ζi : ζ < θ and for every ε < ζ we have ϕε(x,a¯ε) ∈ ∆(N) and N |=(∃x)( ϕ (x,a¯ ))} V ε ε ε<ζ (c) ≤r= being the initial segment relation on Tr (d) M = Mr is the model with universe Tr ∪|N|; without loss of generality Tr∩|N|=∅, with the relations and functions of N,Tr,≤r and • PM =|N| • cM =hi∈Tr • RM = {(b,t) : a ∈ N,t = hϕ (x,a¯ ) : ε < ζ i ∈ T and N |= t,ε t,ε t ϕ (b,a¯ ) for every ε<ζ } t,ℓ t,ε t • FM as in Definition 1.2(j) ϕ,2 • if ι=1 then FM is as in Definition 1.2(i)+. ϕ,1 Remark 1.6. If we adopt 1.3 it is natural to add: (e) for ι = 1,SM = {(ϕ¯1,ϕ¯2) : ϕ¯2 = ϕ1ˆhϕ(x,a¯)i ∈ Tr for some ϕ(x,a¯) ∈ ∆(N)}. Proof. Obvious. (cid:3)1.5 Main Claim 1.7. 1) Assume r is an RSP. If κ = min{tr,pr} then the model N is (κ,1,∆r)-saturated, i.e. ⊕ if p(x) ⊆ ∆r(Nr) is finitely satisfiable in Nr (= is a type in Nr) of cardinality <κ then p is realized in Nr. 10 SAHARONSHELAH 2)Ifθ >ℵ0 andrisaθ-RSP,then Nr is(κ,1,∆r)-saturatedwhereκ=min{tr,pr} recalling 1.1(6), i.e. pr =pTr,θ. 3) If θ > ℵ0,r is a θ-RSP satisfying (k)+ below then Nr is (tr,1,∆r)-saturated when: (k)+ in (T,≤T) any increasing chain which has an upper bound, has a ≤T-lub. Proof. This is an abstract version of [Sh:a, Ch.VI,2.6] = [Sh:c, Ch.VI,2.6]; recall that [Sh:a, Ch.VI,2.7] translates trees to linear orders. 1) Let N =Nr,∆=∆r, etc. Let p be a (∆,1)-type in N of cardinality < κ. Without loss of generality p is infinite and closed under conjunctions. So let (∗) α <κ,p={ϕ (x,a¯ ):α<α }⊆∆(N),p is finitely satisfiable in N. 1 ∗ α α ∗ We shall try to choose t by induction on α≤α such that α ∗ (∗)2 (a) tα ∈T and β <α⇒tβ ≤T tα (b) if β <α then there is b∈RM such that N |=ϕ [b,a¯ ] ∗ tα β β (c) if β <α then b∈RM ⇒N |=ϕ [b,a¯ ]. tα β α If we succeed, this is enough because if t = t is well defined then RM 6= ∅ by α∗ t Definition1.2(h)andanyb∈RM realizesthetypeby(∗) (c)andDefinition1.2(h). t 2 Why can we carry the definition? Case 1: α=0 asLtαet∈tαTr=arntdTt,hheerneciesRntMoαβ=<|Nα|.bAylsDoecfilnaiutsioen(b1).2o(ff)(.∗N)2ohwocldlasubseec(aau)soefp(∗is)2ahtoylpdes and RrMt(T) =|Nr| by Definition 1.2(h). Lastly, clause (c) of (∗) holds trivially. 2 Case 2: α=β+1 If ι = 1 let t = FM (t ,a¯ ) and see clause (i)+ of Definition 1.2. If ι = 2 use ϕβ,1 β β clause (i) of the definition recalling p is closed under conjunctions. Case 3: α a limit ordinal As tTr ≥ κ > α∗ by the claim’s assumption (on tTr, see Definition 1.1(2)) necessarily there is s ∈ T such that β < α ⇒ tα ≤T s. We now try to choose si by induction on i≤α such that ∗ (∗) (a) s ∈T 2.1 i (b) β <α⇒tβ ≤T si (c) j

Atomic saturation of reduced powers

Similar Atomic saturation of reduced powers