Combinatorics of ultrafilters on Cohen and random algebras
aa r X i v : . [ m a t h . L O ] S e p COMBINATORICS OF ULTRAFILTERS ON COHEN ANDRANDOM ALGEBRAS
JÖRG BRENDLE AND FRANCESCO PARENTE
Abstract.
We investigate the structure of ultrafilters on Boolean algebras inthe framework of Tukey reducibility. In particular, this paper provides severaltechniques to construct ultrafilters which are not Tukey maximal. Further-more, we connect this analysis with a cardinal invariant of Boolean algebras,the ultrafilter number, and prove consistency results concerning its possiblevalues on Cohen and random algebras. Introduction
Combinatorial properties of non-principal ultrafilters over ω , that is, ultrafilterson the Boolean algebra P ( ω ) / fin , have been extensively studied for the past halfcentury. Key questions centred around the existence of ultrafilters with additionalproperties such as P -points, the Rudin-Keisler ordering, or cardinal invariants re-lated to ultrafilters. Much less is known about ultrafilters on general Booleanalgebras, although in recent years there has been considerable interest in their rolein set theory and model theory. In particular, Malliaris and Shelah [16] developed anew approach to Keisler’s order based on morality of ultrafilters on Boolean algeb-ras, whereas Goldstern, Kellner and Shelah [9] used Boolean ultrapowers of forcingiterations to force all cardinals in Cichoń’s diagram to be pairwise different.In view of this situation, we investigate combinatorial aspects of ultrafilters oncomplete Boolean algebras, with a particular focus on the following two closelyrelated topics: • the existence of ultrafilters which are not Tukey maximal; • the ultrafilter number.Our results are mostly (but not exclusively) about Cohen and random algebras.Section 2 begins by introducing the central notions of this paper. We thenfollow Kunen’s framework [15] of index-invariant ideals to give examples of Booleanalgebras on which every ultrafilter is Tukey maximal, answering an open questionposed by Brown and Dobrinen [2].In Section 3 we discuss a cardinal invariant related to Tukey reducibility, namelythe ultrafilter number of Boolean algebras. In particular, we focus on the ultrafilternumbers of Cohen and random algebras, discussing their relation with previouslystudied cardinal invariants the continuum, and proving new consistency resultsabout them via finite-support iterations.The last two sections, on the other hand, deal with constructions of ultrafilterswhich are not Tukey maximal. More specifically, in Section 4 we construct non-maximal ultrafilters on Cohen algebras under the assumption that d = 2 ℵ , whereasin Section 5 we carry out such a construction on the random algebra, under thestronger assumption of ♦ . The first author is partially supported by Grant-in-Aid for Scientific Research (C) 18K03398,Japan Society for the Promotion of Science. The second author is an International ResearchFellow of the Japan Society for the Promotion of Science. Tukey-maximal ultrafilters
The notion of
Tukey reducibility , to which this section is dedicated, has its originin the work of Tukey on convergence in general topology.
Definition 2.1 (Tukey [25]) . Let (cid:10) D, ≤ D (cid:11) and (cid:10) E, ≤ E (cid:11) be directed sets. We define (cid:10) D, ≤ D (cid:11) ≤ T (cid:10) E, ≤ E (cid:11) if and only if there exist functions f : D → E and g : E → D such that for all d ∈ D and e ∈ E f ( d ) ≤ E e = ⇒ d ≤ D g ( e ) . The next theorem provides a general upper bound for directed sets of cardinality ≤ κ . Theorem 2.2 (Tukey [25, Theorem II-5.1]) . Let κ be a cardinal and h D, ≤i adirected set. If | D | ≤ κ , then h D, ≤i ≤ T (cid:10) [ κ ] < ℵ , ⊆ (cid:11) . Given a directed set h D, ≤i , as usual we let cof( h D, ≤i ) be the minimum cardin-ality of a cofinal subset of D . Furthermore, if D has no maximum, let add( h D, ≤i ) be the minimum cardinality of an unbounded subset of D . Proposition 2.3 (Schmidt [21]) . Let (cid:10) D, ≤ D (cid:11) and (cid:10) E, ≤ E (cid:11) be directed sets. If D has no maximum and (cid:10) D, ≤ D (cid:11) ≤ T (cid:10) E, ≤ E (cid:11) , then add( E ) ≤ add( D ) ≤ cof( D ) ≤ cof( E ) . Isbell [11] then initiated the study of Tukey reducibility of ultrafilters. Indeed,note that if U is an ultrafilter on a Boolean algebra B , then h U, ≥i is a directedset. Motivated by Theorem 2.2, we say that U is Tukey maximal if and only if (cid:10) [ B ] < ℵ , ⊆ (cid:11) ≤ T h U, ≥i . Theorem 2.4 (Isbell [11, Theorem 5.4]) . For every infinite cardinal κ , there existsa Tukey-maximal ultrafilter over κ . The above theorem stimulated a fruitful line of research, which is nicely surveyedby Dobrinen [5] and constitutes the main motivation for this work. The followingcriterion is particularly useful to determine whether an ultrafilter is Tukey maximal.We omit the proof, which is the same as Dobrinen and Todorčević [6, Fact 12].
Proposition 2.5.
Let κ be an infinite cardinal. For an ultrafilter U on a Booleanalgebra B , the following conditions are equivalent: • (cid:10) [ κ ] < ℵ , ⊆ (cid:11) ≤ T h U, ≥i ; • there exists a subset X ⊆ U with | X | = κ such that every infinite Y ⊆ X is unbounded in U . At this point, it is appropriate to discuss the connection with the Rudin-Keislerordering and ultrafilters over ω . Recall that, whenever U and V are ultrafilters over ω , we have U ≤ RK V if there exists a function f : ω → ω such that for all X ⊆ ωX ∈ U ⇐⇒ f − [ X ] ∈ V. Now, as noted by Dobrinen and Todorčević [6, Fact 1], if U ≤ RK V then h U, ⊇i ≤ T h V, ⊇i . We aim to show that this implication is also true, in the Boolean-algebraiccontext, for the generalized Rudin-Keisler ordering introduced by Murakami. Definition 2.6 (Murakami [20]) . Let B and C be complete Boolean algebras. Forultrafilters U ⊂ B and V ⊂ C , we define U ≤ RK V if there exist v ∈ V and acomplete homomorphism f : B → C ↾ v such that U = f − [ V ] . OMBINATORICS OF ULTRAFILTERS ON COHEN AND RANDOM ALGEBRAS 3
Proposition 2.7.
Let B and C be complete Boolean algebras, with ultrafilters U ⊂ B and V ⊂ C . If U ≤ RK V , then h U, ≥i ≤ T h V, ≥i .Proof. Suppose there exist v ∈ V and a complete homomorphism f : B → C ↾ v such that U = f − [ V ] . Let us define g : C −→ B c ^ { b ∈ B | c ≤ f ( b ) } and note that c ≤ f ( b ) implies g ( c ) ≤ b . Furthermore, observe that if c ∈ V then,by completeness of the homomorphism f , f ( g ( c )) = f (cid:16)^ { b ∈ B | c ≤ f ( b ) } (cid:17) = ^ { f ( b ) | c ≤ f ( b ) } ≥ c, hence f ( g ( c )) ∈ V , which means g ( c ) ∈ U . In conclusion, the maps f and g witnessthat h U, ≥i ≤ T h V, ≥i . (cid:3) The following is a straightforward corollary which clarifies the relationship toultrafilters over ω . Corollary 2.8.
Let B be a complete Boolean algebra. For an ultrafilter V ⊂ B , thefollowing conditions are equivalent: (1) V is not σ -complete; (2) there exists a non-principal ultrafilter U over ω such that U ≤ RK V ; (3) there exists a non-principal ultrafilter U over ω such that h U, ⊇i ≤ T h V, ≥i .Proof. (1 = ⇒ Assuming V is not σ -complete, there exists a maximal antichain { a i | i < ω } in B such that for all i < ω , a i / ∈ V . The function f : P ( ω ) −→ B X _ { a i | i ∈ X } is a complete homomorphism, hence U = f − [ V ] is the desired ultrafilter. (2 = ⇒ By Proposition 2.7. (3 = ⇒ Suppose U is a non-principal ultrafilter over ω such that h U, ⊇i ≤ T h V, ≥i . By Proposition 2.3 we have add( V ) ≤ add( U ) = ℵ , implying V is not σ -complete. (cid:3) In the remainder of this section, we introduce the framework of Kunen [15] andshow that certain Boolean algebras have only ultrafilters which are Tukey maximal.
Definition 2.9. A σ -ideal I over ω is index invariant if for every injective function ∆ : ω → ω and X ⊆ ω we have X ∈ I ⇐⇒ { f ∈ ω | f ◦ ∆ ∈ X } ∈ I. Let I be an index-invariant σ -ideal over ω and α an infinite ordinal. We define I ( α ) ⊂ P ( α as follows: X ∈ I ( α ) if and only if there exist an injective function ∆ : ω → α and a set Y ∈ I such that for all f ∈ X , f ◦ ∆ ∈ Y .Prototypical examples of index-invariant σ -ideals are of course the meagre ideal M and the null ideal N , which will be discussed in greater detail from Section 3. Lemma 2.10 (Kunen [15, Lemma 1.5]) . Let I be an index-invariant σ -ideal over ω . For every infinite ordinal α , I ( α ) is indeed a σ -ideal over α .Furthermore, I ( α ) is also “index invariant” in the sense that, for every injectivefunction Γ : α → β and X ⊆ α , X ∈ I ( α ) ⇐⇒ (cid:8) f ∈ β (cid:12)(cid:12) f ◦ Γ ∈ X (cid:9) ∈ I ( β ) . JÖRG BRENDLE AND FRANCESCO PARENTE
Let
Clop( α be the Boolean algebra of clopen subsets of the Cantor space α ,and B ( α be the σ -algebra generated by Clop( α ; note that | Clop( α | = | α | whereas |B ( α | = | α | ℵ . If I is an index-invariant σ -ideal over ω , let(1) B ( I, α ) = B ( α /I ( α ) be the quotient σ -algebra. Remark . By Lemma 2.10, every injective function
Γ : α → β induces a σ -complete embedding Γ ∗ : B ( I, α ) → B ( I, β ) such that if X ∈ B ( α then Γ ∗ (cid:0) [ X ] I ( α ) (cid:1) = (cid:2)(cid:8) f ∈ β (cid:12)(cid:12) f ◦ Γ ∈ X (cid:9)(cid:3) I ( β ) . Theorem 2.12.
Let I be an index-invariant σ -ideal over ω , containing all singletons.For every infinite cardinal κ and every ultrafilter U on B ( I, κ ) , we have (cid:10) [ κ ] < ℵ , ⊆ (cid:11) ≤ T h U, ≥i .Proof. Let U be an ultrafilter on B ( I, κ ) . We define a function x : κ × → Clop( κ as follows: for every α < κ , x ( α,
0) = { f ∈ κ | f ( α ) = 0 } and x ( α,
1) = { f ∈ κ | f ( α ) = 1 } . As U is an ultrafilter, there exists a function g : κ → such that for all α < κ , (cid:2) x (cid:0) α, g ( α ) (cid:1)(cid:3) I ( κ ) ∈ U . Let X = n (cid:2) x (cid:0) α, g ( α ) (cid:1)(cid:3) I ( κ ) (cid:12)(cid:12)(cid:12) α < κ o ; it is sufficient to show that whenever Y ⊆ X is infinite we have V Y / ∈ U , thenconclude using Proposition 2.5.Let ∆ : ω → κ be an arbitrary injective function. Since we are assuming I contains all singletons, we have { g ◦ ∆ } ∈ I . Therefore, by definition, \ n<ω x (cid:0) ∆( n ) , g (∆( n )) (cid:1) = { f ∈ κ | f ◦ ∆ = g ◦ ∆ } = { f ∈ κ | f ◦ ∆ ∈ { g ◦ ∆ } } ∈ I ( κ ) . Now using the fact that I ( κ ) is a σ -ideal, ^ n<ω (cid:2) x (cid:0) ∆( n ) , g (∆( n )) (cid:1)(cid:3) I ( κ ) = " \ n<ω x (cid:0) ∆( n ) , g (∆( n )) (cid:1) I ( κ ) = / ∈ U, as desired. (cid:3) Brown and Dobrinen [2, Question 4.2] asked: if B is an infinite Boolean algebrasuch that all ultrafilters on B are Tukey maximal, is B necessarily a free algebra?Theorem 2.12 provides a negative answer to this question. Indeed it implies that, if κ is a cardinal satisfying κ ℵ = κ and I is, for instance, the meagre (or null) ideal,then every ultrafilter on B ( I, κ ) is Tukey maximal. On the other hand, B ( I, κ ) is a σ -algebra, hence not free by the Gaifman-Hales theorem [7, 10].3. The ultrafilter number
This section is dedicated to the study of a cardinal invariant, the ultrafilternumber, which is closely related to Tukey reducibility. We shall carry out this studyfor Boolean algebras of the form B ( I, κ ) , focusing in particular on the situationwhere κ is either ω or ω , so that the ultrafilter numbers u ( B ( I, ω )) and u ( B ( I, ω )) are indeed two cardinal invariants of the continuum. We begin with standarddefinitions.Let B be an infinite Boolean algebra. We let π ( B ) = min { | D | | D is a dense subalgebra of B } OMBINATORICS OF ULTRAFILTERS ON COHEN AND RANDOM ALGEBRAS 5 be the π -weight of B . Furthermore, we define the ultrafilter number of B as u ( B ) = min { cof( h U, ≥i ) | U ⊂ B is a non-principal ultrafilter } ; for simplicity of notation, let also u = u ( P ( ω )) . Finally, as usual d denotes the dominating number .The main motivation to consider the ultrafilter number in this paper is given bythe following observation, which follows directly from Proposition 2.5. Remark . Let B be an infinite Boolean algebra. If u ( B ) < | B | , then there existsa non-principal ultrafilter on B which is not Tukey maximal.Let N be the ideal over ω consisting of sets which are null with respect to thestandard product measure µ . The cardinal non( N ) , which is the least size of a setwhich is not null, will play a role in our discussion. We also assume some familiaritywith the meagre ideal M , consisting of sets which are countable unions of nowheredense sets.As M and N are both index-invariant σ -ideals, containing all singletons, weshall consider the corresponding quotient algebras as in (1). In accord with usualnotation, for an infinite ordinal α let C α = B ( M , α ) be the Cohen algebra and B α = B ( N , α ) be the Random algebra . Proposition 3.2.
Let α be an infinite ordinal; then both C α and B α are c.c.c.complete Boolean algebras. Furthermore, (1) for every B ∈ B ( α \ M ( α ) there exists a non-empty C ∈ Clop( α suchthat C \ B ∈ M ( α ) ; (2) for every B ∈ B ( α \ N ( α ) and every ε > there exists a non-empty C ∈ Clop( α such that µ ( B ∩ C ) ≥ (1 − ε ) µ ( C ) . We omit the proof of this well-known fact, but we remark that (1) can be foundin Sikorski [22, Theorem 35.1], while (2) follows from Lebesgue’s density theorem.We are ready to undertake the study of u ( C ω ) , u ( C ω ) , u ( B ω ) and u ( B ω ) , begin-ning with a theorem which summarizes the relations between those four cardinalsand other cardinal invariants of the continuum. Theorem 3.3.
The inequalities in the following Hasse diagram hold: ℵ u ( C ω ) ✇✇✇✇✇✇✇✇✇ u ( B ω ) ❍❍❍❍❍❍❍❍❍ u ( C ω ) u ( B ω )cof( N ) u ✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇ d ✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹ ✇✇✇✇✇✇✇✇✇✇ non( N ) ℵ ●●●●●●●●●● ✈✈✈✈✈✈✈✈✈ Proof.
The upper bound u ( C ω )+ u ( B ω ) ≤ ℵ follows directly from the observationthat the Boolean algebras C ω and B ω have both cardinality ℵ . JÖRG BRENDLE AND FRANCESCO PARENTE
As for u ( C ω ) ≤ u ( C ω ) , Remark 2.11 gives a σ -complete embedding f : C ω → C ω which, due to the fact that C ω is c.c.c., is in fact a complete embedding.Consequently, for every ultrafilter U on C ω we have cof (cid:0) f − [ U ] (cid:1) ≤ cof( U ) byProposition 2.3 and Proposition 2.7. The inequality u ( B ω ) ≤ u ( B ω ) is provedanalogously.To prove u ≤ u ( C ω ) , it is sufficient to note that no ultrafilter on C ω is σ -complete,then apply Proposition 2.3 and Corollary 2.8. Similarly for u ≤ u ( B ω ) .We now prove the inequality d ≤ u ( C ω ) . Let U be an ultrafilter on C ω ; since U is not σ -complete, there exists a maximal antichain { a i | i < ω } such that for all i < ω , a i / ∈ U . Let X ⊂ U be such that | X | < d , we aim to prove that X is notcofinal in h U, ≥i . Whenever s ∈ <ω , we let N s = { f ∈ ω | s ⊂ f } be the corresponding element of Clop( ω . By point (1) of Proposition 3.2, forevery b ∈ C ω \ { } there exists some s ∈ <ω such that [ N s ] M ≤ b . Using this fact,for each x ∈ X we may define a function f x : ω → ω as follows. Let i < ω ; as everyelement of U meets infinitely many members of { a i | i < ω } , let j i be the least j ≥ i such that a j ∧ x > . Then we define f x ( i ) = min { n < ω | there exists s ∈ n such that [ N s ] M ≤ a j i ∧ x } . Since | X | < d , there exists an increasing function g : ω → ω such that for all x ∈ X we have g (cid:2) f x . Now define u = _ i<ω (cid:0) a i ∧ [ { f ∈ ω | f ( g ( i )) = 0 } ] M (cid:1) and assume u ∈ U ; the other case when ¬ u ∈ U is completely analogous. Towards acontradiction suppose now X is cofinal in h U, ≥i , this implies the existence of some x ∈ X such that x ≤ u . Since g is not dominated by f x , we can find some i < ω such that f x ( i ) < g ( i ) ; then, by definition of f x there exists some s : f x ( i ) → suchthat [ N s ] M ≤ a j i ∧ x . Putting everything together, we deduce [ N s ] M ≤ a j i ∧ x ≤ a j i ∧ u ≤ [ { f ∈ ω | f ( g ( j i )) = 0 } ] M , which is a contradiction because f x ( i ) < g ( j i ) . This shows that X cannot be cofinalin the ultrafilter and completes the proof of d ≤ u ( C ω ) .The fact that cof( N ) ≤ u ( B ω ) is a consequence of the work of Burke [3]. Moreprecisely, in Case 1 of [3, Theorem 1], he showed the following: if X ⊆ B ω \ { } has the property that for all b ∈ B ω there exists x ∈ X such that either x ≤ b or x ∧ b = , then there exists a cofinal subset Y ⊆ N such that | X | = | Y | . Therefore,the inequality cof( N ) ≤ u ( B ω ) follows from the observation that, whenever U is anultrafilter on B ω and X ⊆ U is cofinal, clearly X satisfies the assumption of Burke’sresult.The other inequalities in the diagram are well known. (cid:3) Regarding the inequalities in the above theorem, we still do not know whetherit is true that u ( C ω ) = u ( C ω ) and u ( B ω ) = u ( B ω ) in ZFC . We now present twoconsistency results, for which we shall assume some familiarity with finite-supportiteration of c.c.c. forcing notions.
Definition 3.4.
A notion of forcing P is σ -centred if P = S n<ω P n where, for all n < ω , every finite subset of P n has a lower bound in P .It is well known that a finite-support iteration of σ -centred forcing notions isalso σ -centred; see Tall [24] for further details on this topic. Theorem 3.5.
It is consistent that u ( C ω ) < non( N ) . OMBINATORICS OF ULTRAFILTERS ON COHEN AND RANDOM ALGEBRAS 7
Towards a proof, we first introduce a notion of forcing inspired by Mathias [18,Definition 1.0] and discuss its basic features. Let { a i | i < ω } be a maximalantichain in C ω , which will be fixed throughout the construction. For α < ω , letalso D α ⊆ C α be a countable dense subalgebra. As previously observed, whenever α ≤ β we have naturally a complete embedding C α −→ C β , hence without lossof generality we may represent C ω as an increasing union of subalgebras: C ω = S α<ω C α .For every infinite α < ω and every ultrafilter U on C α with the property thatfor all i < ω , a i / ∈ U , we define a notion of forcing M ( U, C α ) . Forcing conditionsare pairs p = h s p , S p i , where: • s p ∈ C α ; • for all i < ω , a i ∧ s p ∈ D α ; • for all but finitely many i < ω , a i ∧ s p = ; • S p ∈ U .The ordering is defined as follows: for p, q ∈ M ( U, C α ) , by definition q ≤ α p ⇐⇒ s p ≤ s q and S q ≤ S p and s q ∧ ¬ s p ≤ S p . It is now straightforward to check that h M ( U, C α ) , ≤ α i is a partially ordered set. Lemma 3.6.
The notion of forcing M ( U, C α ) is σ -centred.Proof. As the subalgebra D α is countable, there are only countably many s ∈ C α such that { a i ∧ s | i < ω } ⊆ D α and the set { i < ω | a i ∧ s > } is finite. For everysuch s , let M s = { p ∈ M ( U, C α ) | s p = s } , so that M ( U, C α ) is the countable unionof the M s . Furthermore, finitely many conditions p , . . . , p n ∈ M s are compatible,as witnessed by the stronger condition h s, S p ∧ · · · ∧ S p n i . (cid:3) For an M ( U, C α ) -generic filter G , let g = _ { s p | p ∈ G } ; we then have the following:(1) { i < ω | a i ∧ g > } is infinite, but(2) for each S ∈ U , { i < ω | a i ∧ g ∧ ¬ S > } is finite.The proof relies on standard density arguments. Indeed, to establish Property (1)it is sufficient to prove that for every n < ω the set D n = { p ∈ M ( U, C α ) | |{ i < ω | a i ∧ s p > }| ≥ n } is dense in M ( U, C α ) . So let p be an arbitrary condition; since S p ∈ U , we have that I = { i < ω | a i ∧ S p > } is infinite, so it is possible to find a strictly increasingsequence h i k | k < n i within I . By density of D α , for every k < n let d k ∈ D α besuch that d k ≤ a i k ∧ S p . Then it is clear that h s p ∨ d ∨ · · · ∨ d n − , S p i is a conditionin D n which is stronger than p .As for Property (2), given S ∈ U the set { p ∈ M ( U, C α ) | S p ≤ S } is dense. Therefore, there exists p ∈ G such that S p ≤ S . For every q ∈ G , thereexists a condition r which is stronger than both p and q , hence s q ∧ ¬ s p ≤ s r ∧ ¬ s p ≤ S p ≤ S. As q ∈ G is arbitrary, we deduce g ∧ ¬ S ≤ s p which concludes the proof.In the forcing extension V [ G ] , the set U ∪{ g } has the finite intersection property.Furthermore, if U ′ is any ultrafilter on C α +1 extending U ∪ { g } , then U ′ also hasthe property that for each i < ω , a i / ∈ U ′ . We now turn to the main lemma towardsthe proof of Theorem 3.5. JÖRG BRENDLE AND FRANCESCO PARENTE
Lemma 3.7.
It is consistent that u ( C ω ) < ℵ .Proof. Let us assume V satisfies ℵ < ℵ . We proceed with a finite-support itera-tion h P α , ˙ Q α | α < ω i of the forcing introduced above; alongside the iteration, foreach α < ω we construct a P α -name ˙ U α for an ultrafilter on C ω + α . As the iterationis σ -centred, in particular the c.c.c. will guarantee that cardinals are preserved inthe forcing extension.First of all, let P = {∅} and let ˙ U be the name for any ground-model ultrafilter U on C ω with the property that for all i < ω , a i / ∈ U . For α < ω , suppose P α and ˙ U α are given; we let P α +1 = P α ∗ ˙ M ( U α , C ω + α ) and ˙ g α be a P α +1 -name for the element of C ω + α added generically at this stage. Wethen take ˙ U α +1 to be the name for an ultrafilter on C ω + α +1 extending ˙ U α ∪ { ˙ g α } .Finally, at limit ordinals δ < ω , we simply let ˙ U δ be a name for an ultrafilter on C ω + δ extending S α<δ ˙ U α .In the resulting forcing extension V [ G ] , let U = S α<ω U α . We claim that U isan ultrafilter on C ω ; indeed, if b ∈ C ω then there exists some α < ω such that b ∈ C α . Hence, at some later stage of the iteration, either b or ¬ b is decided to bein U . In conclusion, U is an ultrafilter on C ω such that n g α ∧ _ { a i | i ≥ n } (cid:12)(cid:12)(cid:12) α < ω , n < ω o is cofinal in h U, ≥i , thus witnessing the fact that ℵ = u ( C ω ) < ℵ in V [ G ] . (cid:3) Proof of Theorem 3.5.
Let us assume V satisfies Martin’s Axiom [17] and considerthe forcing of Lemma 3.7, which resulted in a model of u ( C ω ) < ℵ . The forcingis σ -centred, being an iteration of σ -centred forcings, and it is known that suchextension will also satisfy non( N ) = 2 ℵ . Indeed, this is explained in detail in theproof of [1, Corollary 41], which is enough to conclude the proof of our theorem. (cid:3) We have thus established Theorem 3.5 which, together with the inequality non( N ) ≤ u ( B ω ) discussed in Theorem 3.3, also yields the consistency of u ( C ω ) < u ( B ω ) . Wenow turn to the second consistency result. Theorem 3.8.
It is consistent that u ( B ω ) < ℵ . Our proof employs the finite-support iteration of a forcing notion originally in-troduced by Kunen [14]. For every infinite α < ω and every ultrafilter U on B α ,let K ( U, B α ) be the notion of forcing described as follows. Conditions are functions p : ω → U , where • for all n < ω , p ( n + 1) ≤ p ( n ) ; • p is eventually constant; • for all n < ω , µ ( p ( n )) > n +1 .The order relation is pointwise: for p, q ∈ K ( U, B α ) , by definition q ≤ α p ⇐⇒ ∀ n ( q ( n ) ≤ p ( n )) . The following lemma, although originally stated as a consequence of Martin’sAxiom, has a straightforward translation to the present context.
Lemma 3.9 (Kunen [14, Lemma 4.3.1]) . The notion of forcing K ( U, B α ) is c.c.c.Furthermore, if G is a K ( U, B α ) -generic filter and g : ω → B α is defined by g ( n ) = ^ { p ( n ) | p ∈ G } , then the following hold: • for all n < ω , g ( n + 1) ≤ g ( n ) ; OMBINATORICS OF ULTRAFILTERS ON COHEN AND RANDOM ALGEBRAS 9 • for all n < ω , µ ( g ( n )) ≥ n +1 ; • for every u ∈ U there exists n < ω such that g ( n ) ≤ u . Consequently, in the forcing extension V [ G ] , the set U ∪ { g ( n ) | n < ω } has thefinite intersection property. Proof of Theorem 3.8.
Let us assume V satisfies ℵ < ℵ . As in the proof ofTheorem 3.5, we proceed with a finite-support iteration h P α , ˙ Q α | α < ω i of thec.c.c. forcing just introduced; alongside the iteration, for each α < ω we constructa P α -name ˙ U α for an ultrafilter on B ω + α .The base case of the iteration presents no complications. For α < ω , suppose P α and ˙ U α are given; we let P α +1 = P α ∗ ˙ K ( U α , B ω + α ) and ˙ g α be a P α +1 -name for the function ω −→ B ω + α added generically at thisstage. We then take ˙ U α +1 to be the name for an ultrafilter on B ω + α +1 extending ˙ U α ∪ { ˙ g α ( n ) | n < ω } . Finally, at countable limit ordinals, we proceed as usual bytaking unions.In the resulting forcing extension V [ G ] , let U = S α<ω U α . Then, U is anultrafilter on B ω such that { g α ( n ) | α < ω , n < ω } is cofinal in h U, ≥i , thus witnessing the fact that ℵ = u ( B ω ) < ℵ in V [ G ] . (cid:3) The following question remains open:
Question 3.10.
Is it consistent that u ( B ω ) < u ( C ω ) ?4. Non-maximal ultrafilters on Cohen algebras
Usual constructions of non-maximal ultrafilters over ω , as in Dobrinen and To-dorčević [6, Section 3], rely on the existence of P -points. Definition 4.1 (Gillman and Henriksen [8]) . An ultrafilter U over ω is a P -point if whenever { X n | n < ω } ⊂ U , there exists Y ∈ U such that for each n < ω , Y \ X n is finite.However, if B is an atomless c.c.c. Boolean algebra, then there are no P -points inthe Stone space of B , therefore a different idea is required to obtain ultrafilters on B which are not Tukey maximal. In this section, we present a construction underthe assumption that d = 2 ℵ . Definition 4.2 (Starý [23, Definition 3.1]) . Let B be a complete c.c.c. Booleanalgebra. An ultrafilter U on B is a coherent P -ultrafilter if for every maximalantichain { p i | i < ω } in B , the set n X ⊆ ω (cid:12)(cid:12)(cid:12) _ { p i | i ∈ X } ∈ U o is a P -point ultrafilter over ω .We begin with a reformulation which will be useful for the proof of Theorem 4.6. Lemma 4.3.
Let B be a complete c.c.c. Boolean algebra. For an ultrafilter U on B , the following conditions are equivalent: (1) U is a coherent P -ultrafilter; (2) for every maximal antichain { a i | i < ω } in B and every { x n | n < ω } ⊆ U , there exists y ∈ U such that for each n < ω the set { i < ω | a i ∧ y ∧ ¬ x n > } is finite. Proof. (1 = ⇒ Suppose U is a coherent P -ultrafilter; let a maximal antichain { a i | i < ω } and a subset { x n | n < ω } ⊆ U be given. We aim to find first amaximal antichain { p i | i < ω } such that • for all i < ω there exists j < ω such that p i ≤ a j ; • for every n < ω the set { i < ω | p i ∧ x n > and p i ∧ ¬ x n > } is finite.To do so, it is convenient to use the following notation: for n < ω , let x n = x n and x n = ¬ x n . Now, for s ∈ <ω we define b s = a dom( s ) ∧ ^n x s ( n ) n (cid:12)(cid:12)(cid:12) n < dom( s ) o . First, it is clear that for all s, t ∈ <ω , if b s ∧ b t > then s = t . Secondly, byfinite distributivity we have for each i < ωa i = _ { b s | dom( s ) = i } and consequently _(cid:8) b s (cid:12)(cid:12) s ∈ <ω (cid:9) = _n _ { b s | dom( s ) = i } (cid:12)(cid:12)(cid:12) i < ω o = _ { a i | i < ω } = . Thirdly, we observe that for every n < ω (cid:8) s ∈ <ω (cid:12)(cid:12) b s ∧ x n > and b s ∧ ¬ x n > (cid:9) ⊆ (cid:8) s ∈ <ω (cid:12)(cid:12) dom( s ) ≤ n (cid:9) and the set on the right-hand side is finite. In conclusion, if we enumerate { p i | i < ω } = (cid:8) b s (cid:12)(cid:12) s ∈ <ω (cid:9) \ { } , then it is clear that { p i | i < ω } is a maximal antichain in B satisfying the twodesired properties.For every n < ω , the set X n = { i < ω | p i ∧ x n > } is such that W { p i | i ∈ X n } ∈ U hence, by the assumption that U is a coherent P -ultrafilter, we can find Y ⊆ ω such that y = W { p i | i ∈ Y } ∈ U and for each n < ω , Y \ X n is finite. Now weobserve that for n < ω { i < ω | p i ∧ y ∧ ¬ x n > } = { i ∈ Y | p i ∧ x n > and p i ∧ ¬ x n > } ∪ ( Y \ X n ) which is finite. From this and the fact that { p i | i < ω } refines { a i | i < ω } , itfollows that for each n < ω also { i < ω | a i ∧ y ∧ ¬ x n > } is finite. (2 = ⇒ Let { a i | i < ω } be a maximal antichain in B . Assume that for each n < ω , X n ⊆ ω is such that x n = W { a i | i ∈ X n } ∈ U . By our hypothesis, thereexists y ∈ U such that for each n < ω the set { i < ω | a i ∧ y ∧ ¬ x n > } is finite.Letting Y = { i < ω | a i ∧ y > } , it is obvious that W { a i | i ∈ Y } ∈ U and foreach n < ω , Y \ X n is finite. (cid:3) We note that general existence results for coherent P -ultrafilters are proved inStarý [23, Section 3]; however, the following is all we need for the purpose of thissection. Proposition 4.4 (Starý [23, Proposition 3.4]) . Assume d = 2 ℵ . Let B be acomplete c.c.c. Boolean algebra of cardinality ℵ . Then there exists a non-principalcoherent P -ultrafilter on B . An important observation is that, as opposed to P -points, coherent P -ultrafiltersmay be Tukey maximal. For example, on C ℵ there exists a coherent P -ultrafilterby Proposition 4.4, which is necessarily Tukey maximal by Theorem 2.12. OMBINATORICS OF ULTRAFILTERS ON COHEN AND RANDOM ALGEBRAS 11
Lemma 4.5.
Let B be a complete c.c.c. Boolean algebra, with a dense subalgebra D ⊆ B . Let U be a coherent P -ultrafilter on B and A be a maximal antichain withthe property that A ∩ U = ∅ . Then for all u ∈ U there exists v ∈ U such that v ≤ u and { a ∧ v | a ∈ A } ⊆ D .Proof. Let u ∈ U be arbitrary; by density of D , there exists a maximal antichain { p i | i < ω } ⊂ D such that: • for all i < ω there exists a ∈ A such that p i ≤ a ; • for all i < ω , either p i ≤ u or p i ∧ u = .Using the assumption A ∩ U = ∅ , it follows that for all a ∈ A _ { p i | a ∧ p i = } ∈ U. Now, by the fact that n X ⊆ ω (cid:12)(cid:12)(cid:12) _ { p i | i ∈ X } ∈ U o is a P -point over ω , there exists Y ⊆ ω , with W { p i | i ∈ Y } ∈ U , such that forevery a ∈ A the set { i ∈ Y | a ∧ p i > } is finite.We claim that the element defined as v = u ∧ _ { p i | i ∈ Y } has the desired properties. Clearly v ∈ U and v ≤ u . Furthermore, for each a ∈ A we have a ∧ v = _ { a ∧ u ∧ p i | i ∈ Y } = _ { p i | i ∈ Y and p i ≤ a ∧ u } , but there are only finitely many i ∈ Y such that a ∧ p i > . This means that a ∧ v isa finite (possibly empty) disjunction of elements of the subalgebra D , and therefore a ∧ v ∈ D . (cid:3) Theorem 4.6.
Let B be a complete c.c.c. Boolean algebra. If U is a coherent P -ultrafilter on B , then D(cid:2) π ( B ) + (cid:3) < ℵ , ⊆ E (cid:2) T h U, ≥i Proof.
Let X ⊆ U be such that π ( B ) < | X | ; we aim to find some infinite Y ⊆ X such that V Y ∈ U . If U is σ -complete then this is immediate, so let us assume U is not σ -complete, which means there exists a maximal antichain { a i | i < ω } suchthat for all i < ω , a i / ∈ U .Let D ⊆ B be a dense subalgebra such that | D | = π ( B ) . Without loss of gener-ality, we may assume that for all u ∈ X , { a i ∧ u | i < ω } ⊆ D . Indeed, by Lemma4.5, for all u ∈ X there exists v u ∈ U such that v u ≤ u and { a ∧ v u | a ∈ A } ⊆ D .In case we have chosen the same v u for infinitely many u ∈ X , we have the thesisalready. On the other hand, if the map u v u is finite-to-one, without loss ofgenerality we may replace X with { v u | u ∈ X } , which also has cardinality greaterthan π ( B ) .Now, we claim that there is some x ∈ X such that for all n < ωX n = { u ∈ X | ( ∀ i < n )( a i ∧ x = a i ∧ u ) } is infinite. If not, then for each x ∈ X we could find some n x < ω such that the set(2) { u ∈ X | ( ∀ i < n x )( a i ∧ x = a i ∧ u ) } is finite. Clearly there exists X ′ ⊆ X , with π ( B ) < | X ′ | , such that for all x ∈ X ′ , n x = m for some fixed m < ω . Now, the function X ′ −→ D m x a i ∧ x | i < m i cannot be injective, hence there must be some infinite X ′′ ⊆ X ′ such that x, x ′ ∈ X ′′ implies ( ∀ i < m )( a i ∧ x = a i ∧ x ′ ) , contradicting the assumption that the set in (2)is finite. This completes the proof of the claim.Since for every n < ω the set X n is infinite, recursively we can choose { x n | n <ω } ⊂ X such that for each n < ω , x n ∈ X n \ { x, x , . . . , x n − } . By Lemma 4.3,there exists y ∈ U such that for each n < ω the set { i < ω | a i ∧ y ∧ ¬ x n > } is finite. From now on, we follow the steps of Kanamori [13, Theorem 1.10]. For n < ω we define two natural numbers γ n = max { γ < ω | ( ∀ i < γ )( a i ∧ x = a i ∧ x n ) } and δ n = min { δ < ω | γ n ≤ δ and ( ∀ i ≥ δ )( a i ∧ y ≤ x n ) } . We also define the corresponding interval of natural numbers I n = [ γ n , δ n ) , possibly empty.Note that for each n < ω we have n ≤ γ n , hence we can easily find by recursionan infinite set W ⊆ ω such that for all n, m ∈ W , if I n ∩ I m = ∅ then n = m . Nowthere is clearly an infinite subset Z ⊆ W with the property that z = _( a i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ∈ [ n ∈ Z I n ) / ∈ U. Since U is an ultrafilter, we have x ∧ y ∧ ¬ z ∈ U ; our goal is now to show that forevery n ∈ Z (3) x ∧ y ∧ ¬ z ≤ x n . Let n ∈ Z be given. As { a i | i < ω } is a maximal antichain, to establish (3) it issufficient to prove that for any i < ωa i ∧ x ∧ y ∧ ¬ z ≤ x n . If i is such that a i ≤ z then we are done, so let us assume a i ∧ ¬ z > . Thisimplies that i / ∈ I n , so there are now two possibilities. If i < γ n then by definition a i ∧ x ≤ x n . Otherwise, if δ n ≤ i then again by definition a i ∧ y ≤ x n . Thisestablishes (3) for each n ∈ Z .In conclusion, by taking Y = { x n | n ∈ Z } , it follows that Y is an infinite subsetof X such that x ∧ y ∧ ¬ z ≤ V Y and therefore V Y ∈ U . The thesis now followsfrom Proposition 2.5. (cid:3) Corollary 4.7.
Assume d = 2 ℵ . Let B be a complete c.c.c. Boolean algebra ofcardinality ℵ . If B has a dense subalgebra of cardinality < ℵ , then there existsa non-principal ultrafilter on B which is not Tukey maximal.Proof. By Proposition 4.4 and Theorem 4.6. (cid:3)
The corollary gives that, assuming d = 2 ℵ , for every cardinal ℵ ≤ κ < ℵ there exists an ultrafilter on C κ which is not Tukey maximal. However, this resultdoes not apply to B ω , because d = 2 ℵ implies that all dense subalgebras of B ω have cardinality ℵ (by the fact that d ≤ cof( N ) and [4, Theorem 2.5]). Therefore,in the next section we shall take a different approach. OMBINATORICS OF ULTRAFILTERS ON COHEN AND RANDOM ALGEBRAS 13 Non-maximal ultrafilters on the random algebra
This section complements the previous one by providing the construction of anultrafilter on B ω which is not Tukey maximal, assuming Jensen’s ♦ . Definition 5.1 (Jensen [12]) . Let ♦ be the following principle: there exists asequence h S α | α < ω i such that S α ⊆ α and, for every X ⊆ ω , the set { α <ω | X ∩ α = S α } is stationary.The above principle easily implies ℵ = ℵ and, by [12, Lemma 6.5], is true inthe constructible universe L .The first construction of non-maximal ultrafilters over ω using diamond prin-ciples is due to Milovich [19, Theorem 3.11]. Here we focus on B ω , which for thesake of convenience we identify with B ( ω \ N during the proof of the next the-orem. More explicitly, we shall not distinguish between a set X ∈ B ( ω and itsequivalence class [ X ] N , with the obvious convention that µ ( X ) = µ (cid:0) [ X ] N (cid:1) . Theorem 5.2.
Assuming ♦ , there exists an ultrafilter on B ω which is not Tukeymaximal.Proof. By ♦ , let us fix a sequence h S α | α < ω i such that S α ⊆ α and, for every X ⊆ ω , the set { α < ω | X ∩ α = S α } is stationary. As noted before, the existenceof such a sequence implies ℵ = ℵ , hence we may enumerate B ω = { B α | α < ω } .We proceed to construct recursively a sequence h U α | α < ω i such that, at eachstage α < ω , the set { U β | β ≤ α } ⊂ B ω has the finite intersection property.Let U = . For α < ω , inductively the set { U β | β ≤ α } has the finiteintersection property, so we may choose U α +1 to be either B α or ¬ B α in such away that { U β | β ≤ α + 1 } still has the finite intersection property. Suppose now δ < ω is a limit ordinal; we distinguish two cases:(1) There exists an infinite Y ⊆ S δ such that { U α | α < δ } ∪ (cid:8)T α ∈ Y U α (cid:9) hasthe finite intersection property. Then choose such a Y and define U δ = T α ∈ Y U α .(2) There is no such a Y . Then define U δ = .This completes the recursive construction, whence it follows that U = { U α | α <ω } is an ultrafilter on B ω .Now let X ⊆ ω be uncountable; we aim to find an infinite Y ⊆ X such that T α ∈ Y U α ∈ U , then conclude by Proposition 2.5 that U is not Tukey maximal.By stationarity there exists, for some sufficiently large κ , a countable elementarysubstructure M (cid:22) H κ such that { X, U } ⊂ M , and a limit ordinal δ < ω suchthat X ∩ M = S δ . Now we only need to find an infinite Y ⊆ X ∩ M such that { U α | α < δ } ∪ (cid:8)T α ∈ Y U α (cid:9) has the finite intersection property, as this will implythat, at stage δ , we are in Case (1) of the construction of U .To simplify notation, as δ is countable, we may enumerate all the possible finiteintersections from { U α | α < δ } as { F m | m < ω } . By recursion, we shall constructfor all n < ω a countable ordinal α n ∈ M , an uncountable X n ∈ P ( X ) ∩ M , andnon-empty clopen sets h C n,m | m ≤ n i . At each step n , the following crucialproperty will hold true:( ⋆ n ) ( ∀ m ≤ n )( ∀ α ∈ X n ) µ F m ∩ \ k
14 + 12 n +1 (cid:19) µ ( C m,m ) ≥ µ ( C m,m ) , as can be seen by an elementary calculation. Consequently, for every n ≥ mµ F m ∩ \ k
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