Tukey classes of ultrafilters on omega
Motivated by a question of Isbell, we show that Jensen's Diamond Principle implies there is a non-P-point ultrafilter U on omega such that U, whether ordered by reverse inclusion or reverse inclusion mod finite, is not Tukey equivalent to the finite …
Authors: ** David Milovich **
TUKEY CLASSES OF UL TRAFIL TERS ON ω DA VID MILOVICH Abstract. Motiv ated b y a question of Isb ell, we sho w that ♦ impli es there is a non-P-p oint U ∈ β ω \ ω suc h that neither hU , ⊇i nor hU , ⊇ ∗ i is T uk ey equiv alent to h [ c ] <ω , ⊆i . W e also show that hU , ⊇ ∗ i ≡ T h [ c ] <κ , ⊆i f or some U ∈ β ω \ ω , assuming cf ( κ ) = κ ≤ p = c . W e also prov e t wo negative ZFC results ab out the possibl e T uk ey classes of ultr afilters on ω . 1. Tukey classes Definition 1. 1. A qua siorder is a set with a tr ansitive reflexive r elation (denoted ≤ by default). A qua siorder Q is a κ -dir ected set if every subset of s iz e less than κ has an upp er b ound. W e abbrevia te “ ω -directed” with “directed.” Definition 1.2. The pro duct P × Q of tw o quasiorders P and Q is defined by h p 0 , q 0 i ≤ h p 1 , q 1 i iff p 0 ≤ p 1 and q 0 ≤ q 1 . Definition 1.3. A subset C o f a qua siorder Q is co final if for all q ∈ Q there exists c ∈ C such that q ≤ c . The co finality of Q (written cf ( Q )), is defined a s follows. cf ( Q ) = min {| C | : C cofina l in Q } Definition 1.4 (T ukey [1 2]) . Given directed sets P and Q and a map f : P → Q , we s ay f is a T ukey map, writing f : P ≤ T Q , if the f - ima ge of every unbounded subset o f P is unbdounded in Q . W e say P is T ukey r educible to Q , writing P ≤ T Q , if there is a T ukey map from P to Q . If P ≤ T Q ≤ T P , then we say P and Q are T ukey eq uiv alent and write P ≡ T Q . Prop ositio n 1. 5 (T ukey [1 2]) . A map f : P → Q is T ukey if and only t he f -pr eimage of every b ounde d subset of Q is b ounde d in P . Mor e over, P ≤ T Q if and only if ther e is a map g : Q → P su ch that the image of every c ofinal subset of Q is c ofinal in P . Theorem 1.6 (T ukey [12]) . P ≡ T Q if and only if P and Q or der emb e d as c ofinal subsets of a c ommon thir d dir e cte d set. Mor e over, if P ∩ Q = ∅ , then we may assume the or der emb e ddings ar e identity maps onto a quasior dering of P ∪ Q . The following is a list of basic facts ab out T uk ey r educibility . • P ≤ T Q ⇒ cf ( P ) ≤ cf ( Q ). • F or a ll ordinals α, β , we hav e α ≤ T β ⇔ cf ( α ) = cf ( β ). • P ≤ T P × Q . • P ≤ T R ≥ T Q ⇒ P × Q ≤ T R . Date : July 25, 2008. 2000 Mathematics Subje ct Classific ation. Primary 54D80, 03E04; Secondary 03E35. Key wor ds and phr ases. T uk ey , ultrafilter. Support provided b y an NSF graduate fellowship. 1 2 DA VID MILOVICH • P × P ≡ T P . • P ≤ T h [cf ( P )] <ω , ⊆i . • F or a ll infinite sets A, B , we hav e h [ A ] <ω , ⊆i ≤ T h [ B ] <ω , ⊆i ⇔ | A | ≤ | B | . • Given finitely many or dinals α 0 , . . . , α m − 1 , β 0 , . . . , β n − 1 , we hav e Y i ω , there exist B k +1 ∈ [ B k ] c and n k ∈ T h ζ ( m ) for all m < n . Set ζ ( n ) = k . Since ω ∗ is an F-space (or, more directly , by an easy dia gonalizatio n argument), there exists z ⊆ ω such that x ζ (4 n ) ,p \ x ζ (4 n +2) ,p ⊆ ∗ z and x ζ (4 n +2) ,p \ x ζ (4 n +4) ,p ⊆ ∗ ω \ z for all n < ω . Supp ose z ∈ U . Then there exis t m < ω and h l , r i ∈ ω × Q m such that h l , r i ⊒ z . Cho ose n < ω such that η (4 n + 3 ) = m a nd ζ (4 n + 2) ≥ l . Then choose q ∈ S 4 n +3 such that q ≥ r . Then h ζ (4 n + 2) , q i ⊒ z . Hence, x ζ (4 n +2) ,q ⊒ z ∩ x ζ (4 n +2) ,p ⊒ x ζ (4 n +4) ,p ⊒ h ζ (4 n + 4) , p i . Hence , h ζ (4 n + 4) , p i ⊑ x ζ (4 n +2) ,q ⊑ h ζ (4 n + 3) , s i for some s ∈ Q , which is absurd b eca use ζ is strictly increasing. By 6 DA VID MILOVICH symmetry , w e can also deriv e an absurdity fro m ω \ z ∈ U . Thus, U is not an ultrafilter on ω , which yields our desire d contradiction. The ab ov e result is optimal in the following sense. It is not hard to s how that, for a fixed regula r uncoun ta ble κ , a construction of Br endle and Shela h [2] can be trivially mo dified to yield of a model of ZFC in which some U ∈ ω ∗ satisfies hU , ⊇ ∗ i ≡ T κ × λ for each λ in a n arbitrar y set of regular cardinals exceeding κ . Definition 3. 14. A qua s iorder Q is sa id to b e κ -like if every b ounded subset of Q has size less than κ . Lemma 3.15. Given a qu asior der Q with an unb ounde d c ofinal su bset C , ther e exists a c ofinal subset A of C such that A is | C | -like. Pr o of. Let h c α i α< | C | : | C | ↔ C . F o r each α < | C | , let a α = c β where β is the least γ < | C | s uch that c γ has no upp er b ound in { a δ : δ < α } , provided such a γ exists. If no such γ exists, then α > 0, so we may set a α = a 0 . Then A = { a α : α < | C |} is a s desired. Theorem 3. 16. Supp ose Q is a dir e cte d set that is a c ountable union of ω 1 -dir e cte d sets. Then hU , ⊇ i 6≤ T Q for al l U ∈ ω ∗ satisfying cf (cf ( hU , ⊇i )) > ω . Pr o of. Seek ing a con tr adiction, suppose U ∈ ω ∗ and cf (cf ( hU , ⊇i )) > ω and f : hU , ⊇i ≤ T Q . By Lemma 3.15, U has a co final subset A that is cf ( hU , ⊇i )-like. Since A is co final, f ↾ A is a T ukey map and |A| = cf ( hU , ⊇ i ). Let Q = S n<ω Q n where Q n is ω 1 -directed for all n < ω . Since c f ( |A| ) > ω , ther e exist n < ω and B ∈ [ A ] |A| such that f [ B ] ⊆ Q n . Since A is |A| -like, B is unbounded. Set I = ω \ T B . F or each i ∈ I , cho ose B i ∈ B such tha t i 6∈ B i . Then T i ∈ I B i = T B ; hence, { B i : i ∈ I } is unbounded. But { f ( B i ) : i ∈ I } is a countable subset o f Q n , and ther efore b ounded. This contradicts our assumption that f is T uk ey . Our next theor em is a po sitive consistency result. Its pro of uses Solov ay’s Lemma [9], which we now state in terms of p . Lemma 3.1 7. If A , B ∈ [[ ω ] ω ] < p and | a ∩ T σ | = ω for al l a ∈ A and σ ∈ [ B ] <ω , then B has a pseudointerse ction b su ch t hat | a ∩ b | = ω for al l a ∈ A . Theorem 3.18. Assume p = c . L et ω ≤ cf ( κ ) = κ ≤ c . Then ther e ex ists U ∈ ω ∗ such that hU , ⊇ ∗ i ≡ T h [ c ] <κ , ⊆i . Pr o of. Given a se t E , let I ( E ) denote the set of injections from κ to E . Given E ⊆ P ( ω ), let Φ( E ) denote the se t of h ρ, Γ i ∈ [ E ] <ω × I ( E ) <ω satisfying T ρ ⊆ ∗ S f ∈ ran Γ f ( γ ) for all γ < κ . Let h S α i α< c enum erate [[ ω ] ω ] <κ . Note that if |E | ≥ κ , then Φ( E ) = ∅ implies that E has the SFIP and that hE , ⊇ ∗ i is κ -like. Let us construct a se q uence h U α i α< c in [ ω ] ω such that we have the following for all α ≤ c , given the notation U β = { U γ : γ < β } for all β ≤ c . (1) ∀ β < α ∀ σ , τ ∈ [ U β ] <ω T σ ⊆ ∗ S τ or T σ \ S τ 6⊆ ∗ U β (2) ∀ β < α ∃ σ ∈ [ S β ] <ω U β ∩ T σ = ∗ ∅ or ∀ S ∈ S β U β ⊆ ∗ S (3) Φ( U α ) = ∅ Clearly , (1) and (2) will b e pre served at limit stages of the co ns truction. Let us show that (3) will also be preserved. Let ω ≤ cf ( η ) ≤ η ≤ c and supp o se (1) and (3 ) ho ld for all α < η . Seeking a contradiction, supp ose h ρ, Γ i ∈ Φ( U η ); we may assume h ρ, Γ i is c hosen so a s to minimize dom Γ. By (1), h U α i α<η is TUKEY CLASSE S OF UL TRAFIL TERS ON ω 7 injectiv e; let ψ b e its in verse. Since Φ( U sup( ψ [ ρ ]) ) = ∅ , w e hav e Γ 6 = ∅ . By the pigeonhole principle, there exist A ∈ [ κ ] κ and i ∈ dom Γ such that for all γ ∈ A w e hav e ψ (Γ( i )( γ )) = max j ∈ dom Γ ψ (Γ( j )( γ )). By symmetry , w e may as s ume i = max(dom Γ). Since Φ( U sup( ψ [ ρ ]) ) = ∅ , w e ha ve | A ∩ Γ( i ) − 1 sup( ψ [ ρ ]) | < κ ; hence, we may assume A ∩ Γ( i ) − 1 sup( ψ [ ρ ]) = ∅ . By the definition of Φ( U η ), we hav e T ρ \ S j
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