aa r X i v : . [ m a t h . L O ] M a r WEAK DISTRIBUTIVITY IMPLYINGDISTRIBUTIVITY
DAN HATHAWAY
Abstract.
Let B be a complete Boolean algebra. We show that if λ is an infinite cardinal and B is weakly ( λ ω , ω )-distributive, then B is ( λ, κ is a weakly compact cardinal such that B is weakly (2 κ , κ )-distributive and B is ( α, α < κ , then B is( κ, Introduction
Given sets A and B , A B denotes the set of functions from A to B .In this article, λ and κ will denote ordinals, although usually they canbe assumed to be infinite cardinals. As defined in [6], given λ and κ ,we say that a complete Boolean algebra B is ( λ, κ ) -distributive iff Y α<λ X β<κ u α,β = X f : λ → κ Y α<λ u α,f ( α ) for any h u α,β ∈ B : α < λ, β < κ i . Given maximal antichains A , A ⊆ B , we say that A refines A iff ( ∀ a ∈ A )( ∃ a ∈ A ) a ≤ B a . It isa fact that B is ( λ, κ )-distributive iff each size λ collection of maximalantichains in B each of size κ has a common refinement. There is alsoa useful characterization in terms of forcing (which can be found in [6]as Theorem 15.38): Fact 1.1.
A complete Boolean algebra B is ( λ, κ ) -distributive iff (cid:13) B ( ∀ f : ˇ λ → ˇ κ ) f ∈ ˇ V .
Unfortunately, the definition of weakly distributive varies in the lit-erature (for example [7]). We will use the one given by Jech (see [6]).That is, we say that a complete Boolean algebra B is weakly ( λ, κ ) -distributive iff Y α<λ X β<κ u α,β = X g : λ → κ Y α<λ X β This definition has a natural characterization in terms of forcing. Givena set X and f, g : X → κ , we write f ≤ g iff g everywhere dominates f . That is, ( ∀ x ∈ X ) f ( x ) ≤ g ( x ) . Fact 1.2. A complete Boolean algebra B is weakly ( λ, κ ) -distributiveiff (cid:13) B ( ∀ f : ˇ λ → ˇ κ )( ∃ g : ˇ λ → ˇ κ ) g ∈ ˇ V ∧ f ≤ g. We will show the following: Theorem (A) . Let λ be an infinite cardinal. If B is weakly ( λ ω , ω ) -distributive,then B is ( λ, -distributive. Theorem (B) . Let κ be a weakly compact cardinal. If B is weakly (2 κ , κ ) -distributive and B is ( α, -distributive for each α < κ ,then B is ( κ, -distributive. We will then discuss why Theorem (B) does not hold when we have κ = ω instead of κ being weakly compact, and we will show one way tofix the situation using the tower number. Finally, we use the same ideausing the tower number to prove a variation of Theorem (A) involvingweak ( λ κ , κ )-distributivity for κ > ω .2. Functions from λ ω to ω The proof of the following lemma uses the fact that well-foundednessof trees is absolute. It is crucial, for what follows, that this lemma doesnot require ω λ ⊆ M . See [4] for motivation and discussion. Lemma 2.1. For each A ⊆ λ , there is a function f : ω λ → ω such thatwhenever M is a transitive model of ZF such that λ ∈ M and some g : ( ω λ ) M → ω in M satisfies ( ∀ x ∈ ( ω λ ) M ) f ( x ) ≤ g ( x ) , then A ∈ M .Proof. Fix A ⊆ λ . Define f : ω λ → ω by f ( x ) := ( ∀ n < ω ) x ( n ) A,n + 1 if x ( n ) ∈ A but ( ∀ m < n ) x ( m ) A. EAK DISTRIBUTIVITY IMPLYING DISTRIBUTIVITY 3 Let M be a transitive model of ZF such that λ ∈ M but A M .Suppose, towards a contradiction, that there is some g ∈ M satisfying( ∀ x ∈ ( ω λ ) M ) f ( x ) ≤ g ( x ). Let B be the set B := { t ∈ <ω λ : g ( x ) ≥ | t | for all x in M extending t } . Notice that B ∈ M . Let T ⊆ <ω λ be the set of elements of B all ofwhose initial segments are also in B . Note that T is a tree and T ∈ M .For all a ∈ λ , a ∈ A implies h a i ∈ B . Thus, there must be some a ∈ λ such that a A but h a i ∈ B . If there was not, then A couldbe defined in M by A = { a ∈ λ : h a i ∈ B } .Next, for all a ∈ λ , a ∈ A implies h a , a i ∈ B . Thus, by similarreasoning as before, there must be some a ∈ λ such that a A but h a , a i ∈ B . Continuing like this, we can construct a sequence x ∈ ω λ satisfying ( ∀ n < ω ) x ↾ n ∈ B . Thus, ( ∀ n < ω ) x ↾ n ∈ T , so T is notwell-founded.Since well-foundedness is absolute, there is some path x ′ through T in M . Since ( ∀ n < ω ) x ′ ↾ n ∈ B , we have ( ∀ n < ω ) g ( x ′ ) ≥ n , whichis impossible. (cid:3) This implies the following lemma, whose order of quantifiers is notas powerful, but the functions have the ordinal ( λ ω ) M instead of theset of sequences ( ω λ ) M as their domains: Lemma 2.2. Let M be a transitive model of ZF such that the ordinal λ is in M and ( ω λ ) M can be well-ordered in M . Assume that for each f : ( λ ω ) M → ω there is some g : ( λ ω ) M → ω in M such that f ≤ g .Then P ( λ ) ⊆ M .Proof. Consider any A ∈ P ( λ ). Use the lemma above with A to get˜ f : ω λ → ω such that if ˜ g : ( ω λ ) M → ω is any function in M whichsatisfies(1) ( ∀ x ∈ ( ω λ ) M ) ˜ f ( x ) ≤ ˜ g ( x ) , then A ∈ M . Since ( ω λ ) M can be well-ordered in M , fix a bijection η : ( λ ω ) M → ( ω λ ) M in M . Define f : ( λ ω ) M → ω by f ( α ) := ˜ f ( η ( α )) . DAN HATHAWAY That is, the following diagram commutes:( ω λ ) M ˜ f / / ω ( λ ω ) M . η O O f < < ③③③③③③③③③ By hypothesis, let g : ( λ ω ) M → ω be a function in M which every-where dominates f . Define ˜ g : ( ω λ ) M → ω by˜ g ( x ) := g ( η − ( x )) . We have that ˜ g ∈ M and ˜ g satisfies 1, so by the hypothesis on ˜ f , A ∈ M . (cid:3) We now have the main result of this section: Theorem (A) . Let B be a complete Boolean algebra and λ be an in-finite cardinal. If B is weakly ( λ ω , ω ) -distributive, then B is ( λ, -distributive.Proof. Let µ := λ ω . Assume B is weakly ( µ, ω )-distributive. Force with B . Every f : µ → ω in the extension can be everywhere dominated bysome g : µ → ω in the ground model, so applying the lemma abovein the extension (setting M to be the ground model) tells us that the P ( λ ) of the extension is included in the ground model. Hence, B is( λ, (cid:3) Functions from κ to κ with κ Weakly Compact The first lemma in the previous section was the key to the theoremthere. We have a parallel lemma here which, instead of using theabsoluteness of trees being well-founded, uses the tree property to getsimilar absoluteness. It is important that this lemma does not require κ ⊆ M . By weakly compact, we mean strongly inaccessible and havingthe tree property. Lemma 3.1. For each a ∈ κ , there is a function f : κ → κ such thatwhenever M is a transitive model of ZF such that κ ∈ M , <κ ⊆ M , ( κ M can be well-ordered in M , ( κ is weakly compact ) M , and some g : ( κ M → κ in M satisfies ( ∀ x ∈ ( κ M ) f ( x ) ≤ g ( x ) , then a ∈ M . EAK DISTRIBUTIVITY IMPLYING DISTRIBUTIVITY 5 Proof. Fix a ∈ κ 2. Let f : κ → κ be the function f ( x ) := ( ∀ α < κ ) x ( α ) = a ( α ) ,α + 1 if x ( α ) = a ( α ) but ( ∀ β < α ) x ( β ) = a ( β ) . Let M be an appropriate transitive model of ZF. Suppose g : ( κ M → κ in M satisfies ( ∀ x ∈ ( κ M ) f ( x ) ≤ g ( x ). We will show that a ∈ M .Suppose, towards a contradiction, that a M . Let B := { t ∈ <κ g ( x ) ≥ Dom( t ) for all x in M extending t } . Note that by definition, there cannot be any x ∈ κ M satisfying( ∀ α < κ ) x ↾ α ∈ B because if there was such an x , we would have( ∀ α < κ ) g ( x ) ≥ α , which is impossible. Since B need not be a tree, let T ⊆ <κ B all of whose initial segmentsare also in B . Again, T cannot have a length κ path in M . Note thatfor each α < κ , a ↾ α ∈ B . This is because any x ∈ κ M whichextends a ↾ α differs from a (since a M ), and the smallest γ suchthat x ( γ ) = a ( γ ) must be ≥ α , so g ( x ) ≥ f ( x ) = γ + 1 > γ ≥ α = Dom( a ↾ α ) . Since ( ∀ α < κ ) a ↾ α ∈ B , also ( ∀ α < κ ) a ↾ α ∈ T .Now, B ∈ M (since <κ ⊆ M and g ∈ M ) and so T ∈ M .Since ( ∀ α < κ ) a ↾ α ∈ T , ( T has height κ ) M . Since ( κ is stronglyinaccessible) M , we have ( T is a κ -tree) M . Since ( κ has the tree property) M ,there is a length κ path through T in M , which we said earlier was im-possible. (cid:3) As before, this implies the following lemma, whose order of quanti-fiers is not as powerful, but the functions have the ordinal (2 κ ) M insteadof the set of sequences ( κ M as their domains: Lemma 3.2. Let M be a transitive model of ZF such that the ordinal κ is in M , <κ ⊆ M , ( κ M can be well-ordered in M , and ( κ isweakly compact ) M . Assume that for each f : (2 κ ) M → κ there is some g : (2 κ ) M → κ in M such that f ≤ g . Then P ( κ ) ⊆ M .Proof. The proof is similar to that of Lemma 2.2. (cid:3) As before, the main result of this section follows: Theorem (B) . Let B be a complete Boolean algebra and κ be a weaklycompact cardinal. If B is weakly (2 κ , κ ) -distributive and B is ( α, -distributive for each α < κ , then B is ( κ, -distributive.Proof. This follows from the lemma above just as Theorem (A) followedfrom Lemma 2.2. (cid:3) DAN HATHAWAY The Tower Number One might hope that Theorem (B) holds when κ = ω instead of κ being weakly compact. That is, one might hope that if a com-plete Boolean algebra B is weakly (2 ω , ω )-distributive and ( ω, ω , B could be a Suslin algebra (a Suslin algebrais c.c.c. and therefore is weakly ( λ, ω )-distributive for any λ ). How-ever, if we add the assumption that 1 (cid:13) B ( ω < t ), where we will define t soon, then B is ( ω , ω , ω )-distributivity. As a final twist, we will combineseveral ideas to prove a variation of Theorem (A).Recall that t , the tower number , is the smallest length of a sequence h S α ∈ [ ω ] ω : α < κ i satisfying ( ∀ α < β < κ ) S α ⊇ ∗ S β but there is no S ∈ [ ω ] ω satisfying( ∀ α < κ ) S α ⊇ ∗ S (where S ⊆ ∗ S means S − S is finite). It isnot hard to see that ω ≤ t ≤ ω . See [1] for more on t and relatedcardinals. The following lemma is the key. The idea is borrowed fromFarah in [3], who got the idea from Dordal in [2], who got the idea fromBooth. Lemma 4.1. Let κ be such that ω ≤ κ < t . Let M be a transitivemodel of ZFC such that κ ∈ M and ( ∀ α < κ ) P ( α ) ⊆ M . Then P ( κ ) ⊆ M .Proof. Fix κ and M . Since κ ∈ M and ( ∀ α < κ ) P ( α ) ⊆ M , we have <κ ⊆ M . Let F : <κ → [ ω ] ω be a function in M such that for all t , t ∈ <κ t ⊑ t ⇒ F ( t ) ⊇ ∗ F ( t ), and2) t ⊥ t ⇒ F ( t ) ∩ F ( t ) is finite.Such functions are easy to construct by induction (and the Axoim ofChoice). The construction will not get stuck at a limit stage γ < κ because given t ∈ γ ⊆ M and h F ( t ↾ α ) : α < γ i , since γ < t thereis some S ∈ [ ω ] ω ⊆ M such that ( ∀ α < γ ) S ⊆ ∗ F ( t ↾ α ). The set F ( t ) can be defined to be the least such S accoding to some fixedwell-ordering of [ ω ] ω .Now, consider any a ∈ κ 2. We will show that a ∈ M . The sequence h F ( a ↾ α ) : α < κ i is a ⊇ ∗ -chain (in V) of length κ . Since κ < t , fixsome S ∈ [ ω ] ω satisfying( ∀ α < κ ) S ⊆ ∗ F ( a ↾ α ) . EAK DISTRIBUTIVITY IMPLYING DISTRIBUTIVITY 7 Since P ( ω ) ⊆ M , in particular S ∈ M . Within M , the function F andthe set S can be used together to define a : a = [ { t ∈ <κ S ⊆ ∗ F ( t ) } . (cid:3) By applying the lemma above inductively, we get an improvement: Lemma 4.2. Let κ be such that ω ≤ κ < t . Let M be a transitivemodel of ZFC such that P ( ω ) ⊆ M . Then P ( κ ) ⊆ M . This last lemma is closely related to the fact that 2 κ = 2 ω when-ever κ < t . A proof of this using an argument similar to Lemma 4.1can be found in [1]. Martin’s Axiom (MA) implies t = 2 ω , but theoriginal proof [8] that MA implies 2 κ = 2 ω whenever κ < ω used thealmost disjoint coding poset. We now have the application to completeBoolean algebras: Proposition 4.3. Let κ be an infinite cardinal. Let B be a completeBoolean algebra such that B is ( ω, -distributive and (cid:13) B (ˇ κ < t ) .Then B is ( κ, -distributive.Proof. Apply Lemma 4.2 in the forcing extension with M equal to theground model. (cid:3) Let κ be such that ω ≤ κ < t . Any A ∈ [ ω ] ω can be partitioned into2 ω infinite sets with pairwise finite intersection. Thus, fixing λ ≤ ω ,the function F : <κ → [ ω ] ω in Lemma 4.1 can be replaced by a function F : <κ λ → [ ω ] ω satisfying the same conditions. Slightly modifying theproof of Lemma 4.1, we get that if M is a transitive model of ZFC suchthat λ ∈ M and ( ∀ α < κ ) α λ ⊆ M , then κ λ ⊆ M . Inductively applyingthis fact yields an improvement: Lemma 4.4. Let κ and λ be such that ω ≤ κ < t and λ ≤ ω . Let M be a transitive model of ZFC such that λ ∈ M and ω λ ⊆ M . Then κ λ ⊆ M . Now we may combine Lemma 4.4 with the argument in Lemma 2.1.The case κ = ω of this next lemma is already handled by Lemma 2.1. Lemma 4.5. Let κ and λ be such that ω ≤ κ < t and λ ≤ ω . Foreach A ⊆ λ , there is a function f : κ λ → κ such that whenever M isa transitive model of ZFC such that ω λ ⊆ M (and therefore κ λ ⊆ M )and some g : κ λ → κ in M satisfies f ≤ g , then A ∈ M .Proof. Fix κ , λ , and A . Define f : κ λ → κ by f ( x ) := ( ∀ α < κ ) x ( α ) A,α + 1 if x ( α ) ∈ A but ( ∀ β < α ) x ( β ) A. DAN HATHAWAY This is the analogue of the function f defined in Lemma 2.1. Now fix M and some g : κ λ → κ in M satisfying f ≤ g . Note that 2 ω ∈ M sotherefore κ, λ ∈ M . Let B := { t ∈ <κ λ : g ( x ) ≥ Dom( t ) for all x extending t } . Since <κ λ ∪ { κ, λ, g } ⊆ M , also B ∈ M .Assume towards a contradiction, that A M . Arguing just as inLemma 2.1, there is some x ∈ κ λ satisfying ( ∀ α < κ ) x ↾ α ∈ B . Since κ λ ⊆ M , we have x ∈ M , and in particular x is in the domain of g .We now have ( ∀ α < κ ) g ( x ) ≥ α , which is impossible. (cid:3) Lemma 4.6. Let κ and λ be such that ω ≤ κ < t and λ ≤ ω . Let M bea transitive model of ZFC such that ω λ ⊆ M (and therefore κ λ ⊆ M ).Assume that for each f : ( λ κ ) M → κ there is some g : ( λ κ ) M → κ in M satisfying f ≤ g . Then P ( λ ) ⊆ M .Proof. This follows immediately from the previous lemma. (cid:3) Now follows the theorem: Theorem 4.7. Let B be a complete Boolean algebra. Let κ and λ be such that (cid:13) B (ˇ κ < t ) and (cid:13) B (ˇ λ ≤ ω ) . Assume that B is ( ω, λ ) -distributive and weakly ( λ κ , κ ) -distributive. Then B is ( λ, -distributive. Suslin Algebras and MA ( ω )The theorems in this paper relied on absoluteness results concerningtrees. We can get a counterexample to a generalization of these theo-rems by using a Suslin tree (a tree of height ω such that every branchand antichain is at most countable). Recall the following definition: Definition 5.1. A Suslin algebra is a complete Boolean algebra thatis atomless, ( ω, κ )-distributive for each cardinal κ , and c.c.c.It is a theorem of ZFC that there exists a Suslin algebra iff thereexists a Suslin tree. Furthermore, given a Suslin algebra B , there is aSuslin tree (turned upside down) that completely embeds into B , so B isnot ( ω , κ with ω : Counterexample 5.2. Let B be a Suslin algebra. Then B is weakly (2 ω , ω ) -distributive and B is ( ω, -distributive, but B is not ( ω , -distributive. EAK DISTRIBUTIVITY IMPLYING DISTRIBUTIVITY 9 Proof. The only claim left to be verified is that B is weakly (2 ω , ω )-distributive. In fact, we will show that B is weakly ( λ, ω )-distributivefor all λ . To see why, fix λ and fix a B -name ˙ f such that1 (cid:13) B ˙ f : ˇ λ → ω . Since B has the c.c.c., there are only countably many possible valuesfor a given term in the forcing language. In particular, for each α < λ ,there are only countably many possible values for ˙ f ( ˇ α ). For each α < λ ,let g ( α ) < ω be the supremum of these possible values. We now have1 (cid:13) B ( ∀ α < ˇ λ ) ˙ f ( α ) ≤ ˇ g ( α ) . Since ˙ f was arbitrary, by Fact 1.2 B is weakly ( λ, ω )-distributive. (cid:3) Unfortunately, the counterexample above used a Suslin algebra, whichZFC does not prove exists. In particular, we ask the following: Question 5.3. Is it consistent with ZFC that every complete Booleanalgebra that is both ( ω, κ )-distributive for all κ and weakly ( λ, ω )-distributive for all λ must also be ( ω , ω ). By Proposition 4.3, we only needto worry about those B such that 1 (cid:13) B ( ω = t ). We present anotherresult which shows we do not need to worry about complete Booleanalgebras that satisfy both a strong chain condition and enough weakdistributivity laws. The main idea is the following: if we have a size λ collection C of antichains in B each of size κ ′ , then if B is weakly( λ, κ ′ )-distributive, then there is a maximal antichain A ⊆ B such thatbelow each a ∈ A , each antichain in C has < κ ′ non-zero elements.Assuming also that B is ( ω, | B | )-distributive, we can repeatedly applythis construction countably many times until we produce a maximalantichain B ω such that below each b ′ ∈ B ω , each antichain of B hasonly countably many non-zero elements. That is, B ω will witness that B is “locally c.c.c.”. Then, we use a result of Baumgartner to concludethat since B is locally c.c.c. and ( ω, B is either ( ω , B . If we assumethere are no Suslin trees (which follows from MA( ω )), we get that B must be ( ω , Theorem (D) . Assume there are no Suslin trees. Let B be a completeBoolean algebra such that B is ( ω, | B | ) -distributive, B is κ -c.c. for some κ < ℵ ω , and ( ∀ uncountable κ ′ < κ ) B is weakly ( | B | κ ′ , κ ′ ) -distributive.Then B is ( ω , -distributive. Proof. We will construct a sequence of maximal antichains h B n ⊆ B : n ∈ ω i such that B := { B } and ( ∀ n < m < ω ) B m refines B n . Each B n willhave the property that for any maximal antichain A below an element b ∈ B n , for each b ′ ∈ B n +1 extending b , A will have < | A | non-zeroelements below b ′ . We will then define the maximal antichain B ω torefine each B n , and we will argue that below each b ω ∈ B ω , B is c.c.c.Let κ < ℵ ω be the least cardinal such that B is κ -c.c. Define B := { B } . We will now define a maximal antichain B ⊆ B (which triviallyrefines B ). Every antichain in B has size < κ . Consider an uncountablecardinal κ ′ = ℵ α < κ . Let λ := | B | κ ′ . Let h A β : β < λ i be anenumeration of the maximal antichains in B of size κ ′ . For each β < λ ,let h a β,γ : γ < κ ′ i be an enumeration of the elements of A β . Let ˙ G be the canonical name for the generic filter. Fix a name ˙ f such that1 (cid:13) ˙ f : ˇ λ → ˇ κ ′ and 1 (cid:13) ( ∀ β < ˇ λ ) ˇ a β, ˙ f ( β ) ∈ ˙ G. By hypothesis, B is weakly ( λ, κ ′ )-distributive, so there is a maximalantichain C ,α ⊆ B (which trivially refines B ) and a name ˙ g such that1 (cid:13) ˙ g : ˇ λ → ˇ κ ′ and 1 (cid:13) ( ∀ β < ˇ λ ) ˙ f ( β ) ≤ ˙ g ( β ) . Hence, 1 (cid:13) ( ∀ β < ˇ λ )( ∀ γ < ˇ κ ′ ) γ > ˙ g ( β ) ⇒ ˇ a β,γ ˙ G. This implies that below each c ∈ C ,α , each A β has < | A β | = κ ′ non-zero elements. That is, for each c ∈ C ,α and A β , there are < | A β | many a ∈ A β such that c ∧ a = 0 B .For each ℵ α < κ , we have such a maximal antichain C ,α ⊆ B . Since κ < ℵ ω , the family h C ,α ⊆ B : ℵ α < κ i is countable. Each C ,α has size ≤ | B | , so since B is ( ω, | B | )-distributive, we may fix a singlemaximal antichain B ⊆ B which refines each C ,α . Note that B hasthe property that for each maximal antichain A ⊆ B (below 1 B ) and b ′ ∈ B , A has < | A | non-zero elements below b ′ .We will now define B . Consider an uncountable cardinal κ ′ = ℵ α <κ . Let λ := | B | κ ′ . Let h A β : β < λ i be an enumeration of all size κ ′ antichains that are each a partition of some element of B . Since B isweakly ( λ, κ ′ )-distributive, we may use a similar argument as before toget a maximal antichain C ,α which refines B such that below each c ∈ C ,α , each A β has < | A β | = κ ′ non-zero elements. This completes theconstruction of C ,α . As before, we may use the ( ω, | B | )-distributivityof B to get a common refinement B of every maximal antichain in the EAK DISTRIBUTIVITY IMPLYING DISTRIBUTIVITY 11 family h C ,α : ℵ α < κ i . Note that B has the property that for everypartition A of some element of B and b ′ ∈ B , A has < | A | non-zeroelements below b ′ .We may continue this procedure to get a sequence h B n : n ∈ ω i of maximal antichains of B . The following diagram depicts the maxi-mal antichains which we have constructed, where an arrow representsrefinement: B (cid:15) (cid:15) " " ❊❊❊❊❊❊❊❊ ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ * * ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ C , (cid:15) (cid:15) C , | | ②②②②②②②② C , v v ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ ... t t ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ B (cid:15) (cid:15) " " ❊❊❊❊❊❊❊❊ ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ * * ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ C , (cid:15) (cid:15) C , { { ✇✇✇✇✇✇✇✇✇ C , u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ ... s s ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ ... Using the ( ω, | B | )-distributivity of B once more, we may get a singlemaximal antichain B ω ⊆ B which refines each B n . We will now arguethat given any maximal antichain A ⊆ B and b ω ∈ B ω , A has onlycountably many non-zero elements below b .Fix an arbitrary maximal antichain A ⊆ B . Fix b ω ∈ B ω . Let κ := | A | . If κ ≤ ω , we are done. If not, let b be the uniqueelement of B above b ω . By the construction of B , A has < κ non-zero elements below b . Let κ < κ be the number of such non-zeroelements. That is, letting A := { a ∧ b : a ∈ A } , we have | A | = κ < κ . If κ ≤ ω , we are done because |{ a ∧ b ω : a ∈ A }| ≤ | A | ≤ ω . Otherwise, let b be the unique element of B above b ω . By the construction of B , A has < κ non-zero elementsbelow b . Let κ < κ be the number of such non-zero elements. Thatis, letting A := { a ∧ b : a ∈ A } , we have | A | = κ < κ . If κ ≤ ω , we are done by similar reasonsas before. If not, then we may continue the procedure. However, theprocedure will eventually terminate. This is because if not, then wewould have an infinite sequence of decreasing cardinals κ > κ > κ > ..., which is impossible. Thus, A has only countably many non-zero ele-ments below b ω .At this point, we have argued that below the maximal antichain B ω , B has the c.c.c. Now, it must be that B is ( ω , B is ( ω , B ω . Fix any b ω ∈ B ω . Below b ω , B is c.c.c. and ( ω, B is not ( ω , , there exists a Suslintree which, when turned upside down, can be embedded into B below b ω . However, we assumed there are no Suslin trees. This completes theproof. (cid:3) References [1] A. Blass. Combinatorial cardinal characteristics of the continuum. In M. Fore-man and A. Kanamori, editors, Handbook of Set Theory Volume 1 , pages 395-489. Springer, New York, NY, 2010.[2] P. Dordal. Towers in [ ω ] ω and ω ω . Ann. Pure Appl. Logic , 45:247-276, 1989.[3] I. Farah. OCA and towers in P(N)/fin. Comment. Math. Univ. Carolin. ,37:861-866, 1996.[4] D Hathaway. A lower bound for generalized dominating numbers.arXiv:1401.7948[5] T. Jech. Distributive laws. In R. Bonnet and J.D. Monk, editors, Handbook ofBoolean Algebra . North-Holland Publishing Co., Amsterdam, 1989.[6] T. Jech. Set Theory, The Third Millennium Edition, Revised and Expanded .Springer, New York, NY, 2002.[7] A. Kamburelis. On the weak distributibity game. Ann. Pure Appl. Logic , 66:19-26, 1994.[8] D. Martin and R. Solovay. Internal Cohen Extensions. Ann. Math. Logic , 2:143-178, 1970. Mathematics Department, University of Denver, Denver, CO 80208,U.S.A. E-mail address : [email protected]