Distributivity and Minimality in Perfect Tree Forcings for Singular Cardinals

Dobrinen, Hathaway and Prikry studied a forcing $\mathbb{P}_\kappa$ consisting of perfect trees of height $\lambda$ and width $\kappa$ where $\kappa$ is a singular $\omega$-strong limit of cofinality $\lambda$. They showed that if $\kappa$ is singular of countable cofinality, then $\mathbb{P}_\kappa$ is minimal for $\omega$-sequences assuming that $\kappa$ is a supremum of a sequence of measurable cardinals. We obtain this result without the measurability assumption. Prikry proved that $\mathbb{P}_\kappa$ is $(\omega,\nu)$-distributive for all $\nu<\kappa$ given a singular $\omega$-strong limit cardinal $\kappa$ of countable cofinality, and Dobrinen et al$.$ asked whether this result generalizes if $\kappa$ has uncountable cofinality. We answer their question in the negative by showing that $\mathbb{P}_\kappa$ is not $(\lambda,2)$-distributive if $\kappa$ is a $\lambda$-strong limit of uncountable cofinality $\lambda$ and we obtain the same result for a range of similar forcings, including one that Dobrinen et al$.$ consider that consists of pre-perfect trees. We also show that $\mathbb{P}_\kappa$ in particular is not $(\omega,\cdot,\lambda^+)$-distributive under these assumptions. While developing these ideas, we address natural questions regarding minimality and collapses of cardinals.


Introduction and Background
The study of forcing in set theory is bound to the study of complete Boolean algebras through the fact that every forcing poset can be identified with its regular open algebra.Functions added in the forcing extension correspond to failures of distributivity in the regular open algebra, and subextensions correspond to complete subalgebras.Questions on distributivity and minimality thus motivated some of the early development of forcing [21].Solovay asked the question: For which cardinals κ is there a Boolean algebra that is (ω, ν)-distributive for all ν < κ but not (ω, κ)-distributive?Namba [18] answered this question for regular cardinals κ ≥ ℵ 2 using a perfect tree forcing and an assumption on cardinal exponentiation.In unpublished work, Prikry answered Solovay's question for singular cardinals of countable cofinality, also using a perfect tree forcing.Both forcings are minimal for ω-sequences.More recently, Dobrinen, Hathaway, and Prikry [9] developed this line of investigation further and posed an open-ended version of Solovay's question: Given a regular cardinal λ, for which cardinals κ is there a complete Boolean algebra that is (λ, ν)-distributive for all ν < κ but not (λ, κ)-distributive?This paper focuses on the case where κ is singular and ω < λ = cf(κ) by ruling out many natural candidates.
In pursuing this question, the most natural forcings to consider are higher tree forcings of the sort originally studied by Kanamori using additional requirements on splitting that do not pertain to the countable cases [15].This includes the "vertical" requirement that the trees are both closed under ascending sequences of splitting nodes and the "horizontal" requirement that they split into sufficiently closed filters.Dobrinen et al. point out that the main obstacle to studying higher tree forcings in full generality has to do with the fusion sequences typically used to work with tree forcings-namely, that fusion sequences of length cf(κ) may break down at the first countable step.Our work here considers what happens both with and without Kanamori's splitting requirements.We show that even with the splitting requirements, a variety of perfect tree forcings for singular cardinals κ of uncountable cofinality fail to be (cf(κ), 2)-distributive.This answers a question of Dobrinen et al. regarding the generalization of Prikry's perfect tree forcing.We also show countable distributivity fails for all versions but the ones consisting of trees that obey both the vertical and horizontal requirements.In other words, our findings suggest that failures of countable distributivity predominate in the higher cases when one does not deliberately ensure the viability of fusion sequences.
Most of the introduction to this paper will be a review of definitions of the concepts we are working with.We consider singulars of countable cofinality in Section 2, in which we present a short proof of Prikry's theorem on (ω, ν)-distributivity and obtain ω-minimality for P κ given an ω-strong limit κ of countable cofinality.In Section 3 we move to the case of uncountable cofinality and present a technique whereby, given a perfect tree T of singular width κ and height λ > ω, we consider Bernstein sets on the cofinal branches [T ↾ δ] for some δ < λ to define useful functions λ → On in the extensions.We use this technique to prove that P κ is not (cf(κ), 2)distributive.Then we settle some natural questions on the closed versions of perfect tree forcings and obtain a non-minimality result for one of the versions that is not closed.In Section 4 we define a concept of slow fusion that allows us to work with higher tree forcings that are not as tame as those usually considered in the literature.We use this together with an old trick of Balcar and Simon to obtain various failures of countable distributivity.In Section 5 we obtain failures of three-parameter distributivity.A curiosity resulting from this section is that the regular open algebra of the poset of perfect binary trees in a regular uncountable cardinal κ (a higher perfect tree forcing without Kanamori's restrictions) is isomorphic to that of the Lévy collapse Coll(ω, κ + ) if 2 κ = κ + .Generally, Sections 4 and 5 are relevant to the cases where κ is regular despite this paper's title.
1.1.Distributivity and Boolean Algebras.Here we lay down our most important definitions pertaining to Boolean algebras.More details can be found in Jech's textbook [13] and in the Handbook of Boolean Algebras [12].Definition 1.1.Given a poset P, the regular open algebra ro(P) of P consists of subsets u ⊆ P such that: (i) ∀p ∈ u, q ≤ p, q ∈ u; (ii) ∀p / ∈ u, ∃q ≤ p, ∀r ≤ q, r / ∈ u.
For a Boolean algebra B, we denote by B + = B \ {0 B }.
Fact 1.2.If P is a poset, then ro(P) is a complete Boolean algebra and P embeds densely into ro(P).
• A collection P ⊆ P(B + ) is called a matrix if each member of P is a maximal disjoint subset of B + .• We say that B is (θ, ν, µ)-distributive if for every matrix

Fact 1.4. A forcing poset P is θ-distributive if and only if every function
Definition 1.5.Two posets P 1 and P 2 are called forcing equivalent, denoted P 1 ≃ P 2 , if ro(P 1 ) ∼ = ro(P 2 ).We write P 1 ⋖ P 2 if ro(P 1 ) is isomorphic to a complete subalgebra of ro(P 2 ).1.2.Basic Notions of Tree Forcings.We recall some notions pertaining to trees.Definition 1.6.Let κ be the supremum of an increasing sequence κ = κ α | α < δ of regular infinite cardinals where δ is an ordinal less than or equal to κ.
(1) The set N = N ( κ) = α<δ β<α κ β consists of all functions t such that dom(t) < δ and (∀α ∈ dom(t))(t(α) < κ α ).We typically refer to such functions as nodes.We drop the notation for κ when the sequence is fixed in context.(2) A non-empty subset T ⊆ N that is closed under initial segments is called a tree.(3) For an ordinal α, the set T (α) is the set of t ∈ T with dom(t) = α.(4) The height of a tree T is ht( (1) A non-empty tree T ⊆ N ( κ) is called ever-branching if for each α < δ and for each s ∈ T there is a node t ⊒ s such that closed and ever-branching.(4) A non-empty tree T ⊆ N is called non-stopping (in N ( κ)) if for each node t ∈ T there is a cofinal branch b such that t ∈ b. (5) A non-empty tree T ⊆ N is called pre-perfect (in N ( κ)) if it is everbranching and non-stopping.
The property of being ever-branching is not influenced by the choice of the increasing cofinal sequence κ α | α < δ .We use these notions for various ordinals δ.
Our predominating cardinal arithmetic assumption in the context of perfect tree forcings is as follows: Definition 1.8.We say that κ is a δ-strong limit if for all τ < κ, τ δ < κ.
1.3.Versions of Tree Forcing.We fix κ = κ α | α < λ for a strictly increasing sequence of regular cardinals with ω ≤ λ and let κ = sup( κ).In this paper we would like to cover a wide range of perfect tree forcings P κ involving trees p ⊆ N ( κ) of height λ without being completely encyclopedic.For this purpose, we will cover three main schemes in which tree forcings might vary: (1) The type of tree: Miller versus Laver.Miller-style forcings allow for fairly chaotic branching behavior, whereas Laver-style forcing allows a stem after which all nodes are splitting.(2) The degree of closure of the trees that serve as conditions.This is regarding the closure of the trees themselves as well as the closure of splitting nodes.(3) A specification of the manner of splitting.We consider variations of Condition 1.1, meaning that we will sometimes ask that the splitting sets themselves take a certain form, possibly a club or a stationary set.Now we introduce precise definitions, beginning with the types of trees.Definition 1.9.Let κ = κ α | α < λ be an increasing sequence of cardinals with limit κ and cf(κ) = λ ≥ ω.
Conditions in the forcing M are perfect trees in N ( κ).If the forcing M is defined with respect to a cardinal κ, we may use the notation M κ .
Subtrees are stronger conditions: q ≤ p if q ⊆ p.
Remark 1.10.If κ is any singular cardinal, then the poset M κ is the poset P κ of Dobrinen, Hathaway, and Prikry [9] referred to in the abstract.
We recall a Laver-style relative L κ of M κ to which our results also apply.
Definition 1.11.Let κ = κ α | α < λ be an increasing sequence of regular cardinals that converges to κ, and λ < κ.Singular Laver-Namba forcing L κ is defined as follows: Conditions are (< λ)-closed trees p ⊆ N with a unique stem s such that In L κ subtrees are stronger conditions.
Next, we give definitions regarding the closure of the trees that serve as conditions.Definition 1.12.If conditions are (< λ)-closed trees p, we just write M κ or L κ .If we weaken perfect to pre-perfect, we denote these posets pre M κ or pre L κ .Definition 1.13.If conditions are "(< λ)-splitting-closed" trees p, i.e., any increasing < λ-sequence t α | α < δ , δ < λ, of κ α -splitting nodes t α in p has a limit (the union) which is a κ δ -splitting node in p.We write scl M for the (< λ)-splitting-closed version of M. In the case of Laver forcings (< λ)closure implies (< λ)-splitting closure, so there is no meaningful distinction.
For λ = ω, the "vertical" requirements coincide and describe the trees without maximal nodes.
Finally, we give definitions regarding the manner of splitting.Proof.We show that for a decreasing sequence p α | α < δ of conditions of length δ < λ, the intersection q = α<δ p α is a lower bound.In the Miller case, we have to show that q ∈ scl M club κ .The tree q is not empty, because ∅ ∈ p α for each α < δ.Since any p α is a (< λ)-closed tree, q is also a (< λ)closed tree.Now we show that q is ever-branching.Given s ∈ q, we choose an ❁-increasing sequence t α | α < δ such that t α ∈ Split α (p α ) is above s.We let t δ = {t α | α < δ}.Then for any α < δ, the limit t δ is a (≥ κ δ )splitting node in p α , because p α fulfills that any (< λ)-limit of splitting nodes is a splitting node of the fitting degree.Now we use δ < λ ≤ κ δ and the (< κ δ )-completeness of the club filter in order to intersect OSucc pα (t δ ), α < δ, and find that t δ is a κ δ -splitting node in q.The proof for club Laver forcing L club κ is simpler.
Observation 1.16 is sharp in the case of uncountable λ.We will prove in Section 4 that the other named variations are not ω-distributive.Waiving the "vertical condition" that any (< λ)-limit of splitting nodes is a splitting node (see Subsection 4.2) or waiving the closure of set of ordinal splitting sets (see Subsections 4.3,4.4)allows us to show the failure of ω-distributivity.All pre-perfect versions strongly violate ω-distributivity (see Subsection 5.1).
1.4.Notable Features of Perfect Tree Forcing.A notable feature of perfect tree forcings in general is that of minimality, which motivated Sack's early work [19].For our purposes, we will consider a relaxed version of minimality.
Definition 1.17 ([19]).If λ is a cardinal, a forcing P is minimal for λsequences if for any transitive model In the case of countable cofinality, minimality is often understood through the rendering of the generic branch.This is also justified in the higher cases where we are dealing with (< λ)-closed notions of forcing.Proposition 1.18.Let P be scl M club κ or P = L club .Then the so-called generic branch Proof.The (< λ)-closure of the forcing notion implies that . Then q ≤ p, since otherwise we could pick t ∈ q \ p and have Then there is some q ∈ G such that q g G ∈ [p].We have q ≤ p and thus p ∈ G.
In all of our minimality results, we will work with the generic branch.For λ = ω, the closure requirement is vacuous.For uncountable λ, we will show that the closed versions are non-minimal, as is at least one of the non-closed versions (see Theorem 3.11 below).
Another notable property of tree forcings is that of fusion.
Definition 1.19.We fix a sequence κ α | α < λ of regular cardinals converging to κ.The notions Split α (p) and ≤ α depend on this choice.Let P be any of our forcings.For convenience, we restrict all our forcing notions to the dense sets of conditions in which Stem(p) is κ 0 -splitting.We write λ = cf(κ).
We recall the so-called fusion lemma: Lemma 1.20.Suppose that either cof(κ) = λ = ω and that P is any of the perfect versions or that cof(κ) = λ > ω and that P is one of the closed versions, Considerations of fusion become more complicated when we consider the cases of higher cofinality.In particular, the fusion lemma implies some (< λ)-closure: For this look at a limit δ < λ and let p α | α < δ be given.If q = {p α |α < δ} is a condition, then Split δ (q) = ∅, and for any t ∈ Split δ (q) we have that q ↾ t is a lower bound of p α ↾ t | α < δ ; the latter is just an ordinary ≤-descending sequence.In Section 4 we will explore situations in which closure of the forcing cannot be assumed.

2.1.
A Short Proof of Prikry's Theorem.We begin by giving a proof of Prikry's theorem (see [9,Theorem 3.5]) that harmonizes with the proof from Jech's textbook [13] that under CH the classical Namba forcing is (ω, ω 1 )distributive.However, we do not use a rank function.This approach would also work for the classical Namba forcing.
The following proposition is what allows us to avoid a rank function.Proposition 2.2.Suppose that κ is the supremum of κ = κ α |α < δ where δ is an ordinal less than κ and let T ⊂ N ( κ) be a tree that has no perfect subtree.Then there is a maximal antichain Proof.We choose elements inductively: Suppose t ξ |ξ < η has been defined and the sequence is not already a maximal antichain.Then we can find some t η incomparable with t ξ for ξ < η that has the desired property.Otherwise, the downwards closure of {s ∈ T | ∀ξ < η, t ξ ⊥ s} would be a perfect subtree of T .Now we can prove Theorem 2.1.
Proof.Let p " ḟ : ω → ν".We will find some q ≤ p and g : ω → ν such that q " ḟ = ǧ".We define antichains A n for n < ω by induction on n < ω.At the same time, we define sets {q s : s ∈ n<ω A n } and {α s : s ∈ n<ω A n } by induction on the length of s such that the following hold: (i) All elements of A n lie below elements of The construction goes as follows: Let q ∅ ≤ p be a condition deciding ḟ (0) such that if s = Stem(q ∅ ) then s is κ 0 -splitting.Then let A 0 = {s}.Now suppose A n has been defined.Given s ∈ A n , for each successor t of Stem(q s ), choose q t ≤ q s and α t such that t ⊑ Stem(q t ), q t " ḟ (n + 1) = α t ", and Stem(q t ) is a κ n+1 -splitting node.Then collect all such t's and include them in A n+1 .
Let q be the downwards closure of n<ω A n , which is a perfect tree and hence a condition in M κ .For any g : ω → ν, we define q(g) ≤ q (not necessarily perfect) as follows: Claim: There is some g : ν → ω such that q(g) is a perfect tree.
To prove the claim, suppose otherwise for contradiction.Then for each g : ω → ν, there is an antichain A g ⊂ q given by Proposition 2.2 such that for all t ∈ A g , ∃n t g such that ∀t ′ ⊒ t, t ′ is < κ n t g -splitting.Now construct a ❁-increasing sequence s n | n < ω where s n ∈ A n for each n as follows: Make arbitrary choices until we have κ n > ν ω .Then suppose we are given s n and let t = Stem(q sn ).Then the set Now let g be given by g(n) = α sn for n < ω.There is some t ∈ A g and some n such that s n ⊒ t.But we have a contradiction when we consider s m such that m ≥ n, n t g .Hence we have proved the claim.If g : ν → ω witnesses the claim, then q(g) ∈ M κ and q(g) " ḟ = ǧ.This style of argumentation works for other tree forcings for singulars of countable cofinality.In particular, we obtain a new proof of the same result for the forcing L stat κ studied by Cummings We will use an important aspect of the Cummings-Magidor arguments, which to some extent derives from Laver's original forcing [17].Fact 2.4 (See, e.g., [8, Fact one, page 3341]).If cf(κ) = ω, L stat κ is defined on the product N ( κ), p ∈ L stat κ has a stem of length at least n, and p forces that γ is an ordinal below µ for some µ < κ n , then there is some q ≤ 0 p such that q decides a value for γ.
Proof of Theorem 2.3.Suppose p "f : ω → ν" and use Fact 2.4 to assume without loss of generality that p has a stem of length m where ν ω < κ m and p decides ḟ (n) for all n ≤ m.
We define a a condition q ≤ p as well sets {q s : s ∈ q} and {α s : s ∈ q} with the following properties: This construction is a more controlled analog of the one in the proof of Theorem 2.1, and it works using Fact 2.4.We also define q(g) for g : ω → ν as in the proof of Theorem 2.1.The condition q ≤ p will be the union of all of the s's.
We construct a tree r ≤ q by induction on the length of nodes t as follows: r has the same stem as q, and if we have established that t ∈ r and |t| = n, then include t ⌢ α ∈ r if and only if α is not in the set {β < κ n :∃g : ω → ν, t ∈ q(g) but has a non-stationary set of successors in q(g), and β ∈ OSucc q(g) (t)}, which is non-stationary.
We claim that there is some g such that q(g) ∩ r is a condition in L stat κ .Suppose otherwise.Then for every t ∈ r, t fails to be a stem of q(g)∩r.This means that there is some s ⊒ t such that |s| = n for some n and the set of successors of s in q(g)∩r is non-stationary in κ n , and so Succ (r↾s)∩q(g) (s) = ∅.For each g : ω → ν, we thus build maximal antichains A g ⊆ r using the idea of Proposition 2.2 such that for all s ∈ A g , Succ (r↾s)∩q(g) (s) = ∅.Finally, let b be any cofinal branch of r and let h : n → α b↾n .Then there is some t ∈ b and some s ∈ A h such that s ⊆ t, which is a contradiction because this implies that t / ∈ r.

Minimality Without Measurables.
As before, P is M κ where κ is a singular of countable cofinality.We prove the following theorem, which waives the assumption in Theorem 6.6 from [9] that the κ n 's are measurable.
Here we use Γ(P) to refer to the canonical P-name for a P-generic filter over V .Definition 2.6 (See [9, Definition 6.1] and [5,Lemma 11]).Let P be either M κ or L κ for a singular κ.Let Ȧ be a P-name such that 1 P Ȧ : ω → V ∧ Ȧ ∈ V .For each condition p ∈ P we let ψ p : p → <ω V be the function that assigns to each node t ∈ p the longest sequence s = ψ p (t) such that p ↾ t š ⊑ Ȧ.
The basic idea of Lemma 2.7 is that the generic sequence, and therefore the generic filter, can be reconstructed from ȦG using Proposition 1.18 by determining a condition with the sharp coding.
Thanks to Lemma 2.7, Theorem 2.5 is implied by the following lemma.
Lemma 2.8.Suppose that p Ȧ : ω → V ∧ Ȧ ∈ V .Then there are densely many q ≤ p in M κ that are Ȧ-sharp.Now we state our main lemma.Lemma 2.9.Given q ≤ p, n ∈ ω, and s ∈ q, there is some t ⊒ s and q ′ ≤ q such that Stem(q ′ ) = t, t is κ n -splitting, and t is Ȧ-sharp with respect to q ′ .Proof.The procedure consists of the following five steps: Step 1: We choose t ⊒ s, t ∈ q such that | Succ q (t)| ≥ κ n .
Proof.We use a fusion sequence and the fact that M κ is (ω, ν)-distributive for any ν < κ.
We explain the construction assuming the claim by defining a fusion sequence (p n ) n<ω as follows: Let p 0 = p.Suppose that n ≥ 1 and p n−1 is constructed.We then construct p n such that p n ≤ n−1 p n−1 and such that each Explicitly, for all t ∈ Split n−1 (p n−1 ) and s ∈ Succ p n−1 (t), we find s + ⊒ s and q(t, s) witnessing Claim 2.9.Then we let Finally, we let p be the fusion of (p n ) n<ω so that p witnesses the statement of the lemma.This completes the proof.
Throughout this section, κ is a singular cardinal of uncountable cofinality λ.Let P be any perfect version of M κ or L κ (we exclude the pre-perfect versions here).Theorem 3.1.If cf(κ) = λ > ω and κ is a λ-strong limit, then P is not (λ, 2)-distributive.
We show that our assumptions regarding cardinal arithmetic are sharp using a result that probably goes back to Solovay.

Remark 3.2. If τ, λ < κ and τ
The second line of the statement follows from a coding that we get if λ λ ≥ κ.
We recall the classical result of Bernstein for completeness.
Lemma 3.3 (The Existence of A-Bernstein Sets.).Let µ be an infinite cardinal and let A ⊆ P(µ), be such that |A| ≤ µ and such that for any A ∈ A we have |A| = µ.Then there is a set {B α | α < µ} with the following properties: (1) for each α < µ, B α ⊆ µ and The B α 's are sometimes called Bernstein sets for A.
Proof.We enumerate A as A β | β < µ .By induction on γ < µ we choose an element a β,γ ∈ µ by induction on β < γ such that We will use a simple lemma extending Proposition 2.2 that helps us avoid the assumption of GCH in our proof of Theorem 3.1.Lemma 3.4.Suppose that κ is the supremum of κ = κ α | α < δ where δ is an ordinal less than κ, and that τ δ ≤ κ for all τ < κ.Let T ⊂ N ( κ) be a tree with no branches of length < δ.Then the following are equivalent: (1) T has a perfect subtree.
To prove that (3) implies (1), suppose contrapositively that T does not have a perfect subtree.Let A ⊂ T be the antichain given by Proposition 2.2.Observe that since A ⊂ T and |T | ≤ κ (using that τ δ ≤ κ for all τ < κ), it follows that |A| ≤ κ.If t ∈ A and α is such that ∀t ′ ❂ t, | Succ T (t ′ )| < κ α , it follows that there are no more than κ δ α -many branches containing t, meaning at most κ-many (again using that τ δ ≤ κ for all τ < κ).Therefore, T has at most κ-many branches.
The following can be seen as a variation on Silver's famous theorem for singular cardinals of uncountable cofinality.Lemma 3.5.Let κ be a singular λ-strong limit cardinal of uncountable cofinality λ.Let κ α | α < λ be a sequence of regular cardinals converging to κ, and for α < λ let µ α be the cardinality of β<α κ β .
If T is a perfect tree in the space α<λ κ α , then the set Proof.Let δ α | α < λ be the club such that for all α < λ, δ α+1 = κ α+1 , and such that for all limits α < λ, δ α := sup β<α κ α .There is a club D ⊆ λ such that for all α ∈ D, α is a limit and δ α is a λ-strong limit.Observe that for all α < λ, T ↾ α has no branches of length less than α.Then observe that by Lemma 3.
Therefore, if the conclusion of the lemma we are trying to prove is false, then Now we can demonstrate the basic idea behind our definition of S. For all α ∈ S, enumerate is a cofinal branch, then for α ∈ S let F (α) be the least β < α such that there is some ξ < δ β such that b ↾ α = t α ξ .This is a regressive function on S, so Fodor's Lemma implies that there is a stationary subset S ′ ⊆ S and some β < λ such that for all α ∈ S ′ , b ↾ α = t α ξ for some ξ < δ β .We can use this idea to show that there are at most κ-many cofinal branches of T .Suppose for contradiction that there are at least κ + -many cofinal branches of T .Then because 2 λ < κ, there is a stationary subset S ′ ⊂ S and some β < λ such that are κ + -many branches b with the property that for all α ∈ S ′ , b ↾ α = t α ξ for some ξ < δ β .Then since δ β ≤ κ β and κ λ β < κ, it follows that there is a function h : S ′ → δ β such that for κ + -many distinct branches b, b ↾ α = t α h(α) for all α ∈ S ′ .But these branches are determined cofinally, so this is impossible.
Since T has at most κ-many branches, it cannot be a perfect tree.Now we turn to the proof of Theorem 3.1: Proof.Again let µ α denote | β<α κ β |.For a limit α < λ, let A(α) be the collection of subsets X of β<α κ β of size µ α (i.e. the subsets with "full" size) that are the branches of a perfect tree in γ<α β<γ κ β .Note that there are at most 2 δα = 2 lim β<α κ β = µ α -many perfect trees in γ<α β<γ κ β and hence For α = 0 and successors α, we let B α,γ = {0}.Now we define a P-name for a function from Lim(λ), the set of limit ordinals in λ, to 2. We define ḟ as the following name: Assume for contradiction that there is some g ∈ Lim(λ) 2 ∩ V and there is some p ∈ P such that p ḟ = g.By Lemma 3.5, there is some limit ordinal α > dom(Stem(p)) such that |[p ↾ α]| = µ α , and therefore [p ↾ α] ∈ A(α).If g(α) = 1, then we strengthen p to some condition q such that that Stem(q) ↾ α ∈ B α,0 .This is possible, since B α,0 is a A(α)-Bernstein set and If g(α) = 0, then we strengthen p to some condition q such that Stem(q) ↾ α ∈ B α,0 .This is again possible, since So in any case we have q ≤ p and q ḟ (α) = g(α), and hence a contradiction.Thus, forcing with P adds a new function ḟG ∈ Lim(λ) 2 and hence P is not (λ, 2)-distributive.
This Bernstein argument applies equally to the Laver versions of P (see Def. 1.11( 2)) and to the versions splitting into particular filter sets and also to the versions in which each limit of splitting nodes is a splitting node.Of course, Lemma 3.5 is not necessary for the Laver versions.
The Bernstein technique shows us that Lemma 3.5 leads to a stronger negation of cf(κ)-distributivity: there is a name for a collapsing function.
Proof.For each α < λ we choose a set A(α) and a set {B α,β | β < µ α } of disjoint Bernstein sets as above.We define a P-name for a function from Lim(λ) onto κ: The argument that P " ḟ : Lim(λ) ։ κ" is analogous to the one presented in the proof of Theorem 3.1.For γ < κ the set is a dense subset of P.
Observe that the Bernstein technique does not work for pre-perfect trees.In Section 5.1 we will show failure of (λ, 2)-distributivity for the pre-perfect versions using a different name.

Non-Minimality and Cardinal Preservation for the (< cf(κ))-
Closed Versions.In this section we settle natural questions of minimality and cardinal collapses for scl M club κ and L club κ given singular λ-strong limits of cofinality λ > ω.
The first observation is a corollary to Theorem 3.6.For λ = κ = ω, Groszek [11,Theorem 5] proved an analogous result.For λ = ω < κ, the technique of Brendle et al. [4,Proposition 77] shows that scl M club κ and L club κ add an ω-Cohen real and hence are not minimal for ω-sequences.Note that Theorem 2.5 does not generalize to uncountable λ because we do not have the (λ, ν)-distributivity needed for Step 2 of Claim 2.9.Corollary 3.7.Suppose that P is either scl M club κ or L club κ and that κ is a singular λ-strong limit of cofinality λ > ω.Then Add(λ, 1) ⋖ P, and so P is not minimal for λ-sequences.
Proof.This follows from two folklore lemmas.The first is that for homogenous forcings Q 1 and Q 2 , there is a complete embedding ι : By Theorem 3.6, we have The rest of the section is dedicated to proving that the closed versions preserve κ + as well as the cardinals up to and included λ.
We prove a lemma on ≤ α -stronger decision into boundedly-many possibilities.There are analogs used in many places, e.g.[15].Theorem 3.8.Let p ∈ P force that ḟ be a P-name for a function from λ into κ + .Then there are a set x of size at most κ and a condition q ≤ p such that q range( ḟ ) ⊆ x.
Proof.First we establish a claim that works for any of the ever-splitting versions: force that β is an ordinal and let α < λ.Then there is some q ≤ α p and some set x of size at most | Split α (p)| • κ α such that q β ∈ x.To prove the claim, consider α < κ.By definition, Split α (p) is a maximal antichain.For each s ∈ Split α (p) we proceed as in the step α = 0, i.e., for each t ∈ Succ p (s) we choose some q(t, s) ≤ p ↾ t that decides β and we let x s be a set containing all these decisions.We let x = {x s | s ∈ Split α (p)}.The size of Split α (p) is bounded by sup β<α µ β ≤ κ by our assumptions on cardinal exponentiation in Definitions 1.9 and 1.11.Each x s has size at most Then q ≤ α p and q β ∈ x.This proves the claim.Given p, assume without loss of generality that p ḟ : λ → κ + .Then construct a fusion sequence q α | α < λ such that there are x α of size at most κ such that q α ḟ (α) ∈ x α .We choose the q α be recursion on α.In the successor steps we proceed according to the claim.In the limit steps α < κ, we first let q ′ α = {q β | β < α}.By closure, q ′ α is a condition.Now we perform a successor step to go from q ′ α to q α .The fusion limit q = {q α | α < λ} has the desired properties.
Although the previous lemma applies to the closed version, there is another way of obtaining preservation of κ + for L club κ : Proposition 3.9.The singular club-Laver-Namba forcing L club κ has the κ +chain condition.
Proof.Observe that two conditions in the club-Laver-Namba forcing are compatible if they have the same stem, and there are only κ-many stems.Theorem 3.6 on collapsing κ to λ pertains also to the (< λ)-closed versions of the forcings from this section.Combining Theorem 3.8 with Theorem 3.6 about collapsing κ to λ (for which we used λ-strong limits) we get: Corollary 3.10.For any λ-strong limit κ and P κ being scl M club κ or L club κ , we have V and all cardinals ≤ λ are preserved.

The Non-Miniminality of a
Non-Closed Version.The aim of this section is to show that we do not necessarily need closure to obtain non-minimality results.Theorem 3.11.Let κ be a singular cardinal of cofinality ω 1 .Let P be any of L κ , L stat κ , and L club κ .Then Add(ω 1 , 1) ⋖ P. For the rest of this section we fix κ, a singular cardinal of cofinality ℵ 1 .We will work with L κ for simplicity.Proof.This is the observation that if α is the length of the stem of p, then for all limit ordinals γ in (α, λ), p ↾ γ is perfect and therefore p does not decide ḟ (γ) by the reasoning in the proof of Theorem 3.1.For t ∈ p, we say that t is good if for all β ∈ Lim((dom(t)+1))∩δ, t ↾ β ∈ B β,0 if and only if b( β) = 0 where ω • β = β.
We will show by induction on limits γ ∈ [α + ω, δ] that: for all good nodes s ⊒ s in p, there is some t ⊒ s such that t is good and γ ∈ dom(t) + 1.The successor case of the induction pertains to ordinals of the form γ + ω.Let s ⊒ s be a good node of p such that dom(s) = γ < γ + ω.If γ > α, apply the inductive hypothesis to find s ′ ⊒ s with domain γ.Because p ∈ L κ , (p ↾ s ′ ) ↾ (γ + ω) is perfect.Therefore we can find some t ⊒ s such that the set of predecessors of t is a cofinal branch of (p ↾ s) ↾ (γ + ω) and t ↾ (γ + ω) ∈ B γ+ω,0 if b(γ + 1) = 0 where ω • (γ + 1) = γ + ω and t ↾ (γ + ω) / ∈ B γ+ω,0 otherwise.Then t is as sought.Now suppose that γ is a limit of limit ordinals, and in particular that γ n | n < ω is a sequence of limit ordinals above α converging to γ.We pick a good node s and define a perfect subtree T of (p ↾ s) ↾ γ by induction on n.Suppose we have define T ↾ γ n .Then for each cofinal branch s ′ representing a cofinal branch of T ↾ γ n , choose κ γn+1 -many immediate successors s ′ ξ , ξ < κ γn+1 of s in p (again using that p ∈ L κ ) which are still good nodes, and then choose good nodes s ′′ ξ ⊒ s ξ such that γ n+1 ∈ dom s ′′ ξ for all ξ < κ γn+1 using the inductive hypothesis.Then collect all such s ′′ ξ 's for each s ′ representing a cofinal branch of T ↾ γ n and let this form T ↾ γ n+1 .Once we are done defining T , we choose a node t of length γ + 1 defining a cofinal branch of T such that t ↾ γ ∈ B γ,0 if and only if b(γ) = 0 where ω • γ = γ.
Finally, we choose some t ∈ p witnessing the statement for δ and we let p = p ↾ t.
We proceed with the proof of Theorem 3.11.
The basic properties of complete embeddings hold for ι.We see that ι(b) is a regular open set, meaning that if p / ∈ ι(b), then there is some q ≤ p such that {r ∈ L To prove completeness, suppose that u ∈ ro(L κ ) and let The fact that this construction cannot be generalized from Corollary 3.7 leads us to our next section.

The Balcar-Simon Technique and Failures of Countable Distributivity
Here are some examples of failures of countable distributivity in the nonclosed versions of forcings of perfect tree forcings.We demonstrate the ways in which an idea of Balcar and Simon can be used to obtain failures of countable distributivity by finding a way to degrade a structure through countably many steps (see [1], [2]), mainly by picking out a non-stationary set and discarding limit points at each stage.All of these involve an ordering, and we cover ≺ vert , ≺ horiz , and ≺ stat .For forcing with the pre-perfect versions we refer the reader to Subsection 5.1.
4.1.Slow Fusion.Each of our results in the secution will require a lemma stating that for each condition p in the forcing we are interested in, there is some q ≺ p where ≺ is one of the orders mentioned above.These conditions q will be provided by fusion sequences in which conditions are strengthened locally and only slightly.We call these "slow fusion" sequences.These will in the end be used to prove that the usual sequences in the non-closed versions are not viable because they break down at intermediate steps.
We describe the basic idea of slow fusion here.It is simpler in the Laver case, which by definition satisfies Kanamori's vertical requirement that limits of splitting nodes are splitting nodes.In this case, a ≤-decreasing sequence p α | α < λ is slow if p α differs from {p β | β < α} only in the level p α (dom(Stem(p 0 )) + α + 1), and for any β < α and t ∈ p α (dom(Stem(p 0 )) + α + 1), we have p α ↾ t = p β ↾ t.In other words, only one level is altered at a time.
The Miller case is analogous, but we must keep track of the antichains Split α (p) for α < λ.In this case, in order to slowly descend from {p β | β < α} to p α , a "horizontal stripe" of {p β |β < α} that starts after Split α ( {p β |β < α} and is bounded in the tree ordering is thinned out in the transition to p α , whereas the other areas of {p β | β < α} are just copied into p α .By the disjointness of the stripes shrinkage to a non-condition in an intersection over infinitely many previous steps shall be precluded. The existence of the intermediate limit {p β | β < α} is proved rigorously in Lemma 4.2 below.The final intersection, meaning the eventual limit of the fusion sequence at stage λ, is justified in the same manner.Definition 4.1.We fix a sequence κ α | α < λ of regular cardinals converging to κ.Let P be any of our forcings where λ = cf(κ).
Note that this definition depends on the choice of κ α | α < λ .Lemma 4.2.For all the forcings in Definitions 1.9 and 1.11 and for any slow fusion sequence p α | α < δ with 1 ≤ δ ≤ λ, we have that q δ = {p α | α < δ} ∈ P and that the following equation from the standard fusion lemma holds: where Split ≥δ (p α ) has the natural interpretation.If δ = λ, then the set in the second line is empty.If δ < λ, the right-hand side of the first line is redundant since it is a subset of the set in the second line.
Proof.We show that the fusion does not break down at intermediate steps or at step λ by induction.Specifically, we prove that q δ := α<δ p α ∈ P and that Equation (4.1) holds by induction on δ ≤ λ.We give the argument for the Miller versions.The argument for the Laver versions is simpler because the antichains Split α (p γ ) are levels.
We must show that q δ as defined is a condition.In the case of δ = ε + 1 we have q δ = p ε is a condition and the verification of Equation (4.1) is easy.Now suppose that δ < λ is a limit.Since the empty node is in any p α , we have ∅ ∈ q δ .The main task is then to show that for each node s ∈ q δ , for any ε ∈ [δ, λ), there is a κ ε -splitting node in q δ above t.
By construction we have Split α (p α ) = Split α (q δ ) for α < δ, and for t ∈ Split α (q δ ), Succ pα (t) = Succ q δ (t).Since for g ∈ Split δ (q δ ) p 0 ↾ g = q δ ↾ g, the higher part is a perfect tree, as required.So q δ is indeed a condition and Equation (4.1) holds.Then q δ ≤ δ,α p α for all α < δ and the so condition q δ can serve as a prolongation of the slow fusion sequence p δ | α < δ .
Lemma 4.5.For any p ∈ P κ there is some q ≺ vert p.
Proof.By induction on α we choose a slow fusion sequence p α |α < λ using Lemma 4.3.We let p 0 = p.If α = β +1 is a successor, we let p α = p β .If α is a limit we first let q α = {p β |β < α}.By Lemma 4.2, q α ∈ P κ .Now for any t ∈ Split α (q α ) we pick some f (q α , t) ∈ Succ qα (t) and some g(q α , t) ⊒ f (q α , t) such that g(q α , t) ∈ Split α+1 (q α ).We let Then Split α (p α ) = Split α+1 (q α ) and p β | β ≤ α is an initial segment of a slow fusion sequence.By Lemma 4.2 q = {p α | α < λ} is a condition and for any limit α and any t ∈ Split α (p) ∩ q, | Succ q (t)| = 1 by construction.Thus we have q ≺ vert p. Proof.Let A 0 be any maximal antichain in P κ .Given A n , choose for any p ∈ A n , a maximal antichain of conditions q ≺ vert p and let A n+1 be the union of these q's.Then there is no q ∈ P such that for any n there is p n ∈ A n with q ≤ p n because such a q would have no splitting nodes.(1) p ∈ P. ( (3) For for each α < λ for each s ∈ Split α (p) the function f p,s : OSucc(s) → κ α is one-to-one and strictly increasing.We let L be the set of all labeled trees.Definition 4.9.Let (p, f p ) and (q, f q ) be two labeled trees such that q ≤ p.We write f q < f p if ∀s ∈ Split(q), ∀ξ ∈ dom(f q,s ), f q,s (ξ) < f p,s (ξ).
We write (q, f q ) ≺ horiz (p, f p ) if q ≤ p and f q < f p .Lemma 4.10.Let (p, f p ) be a labeled tree.Then there is some (q, f q ) such that q ≤ 0 p and (q, f q ) ≺ horiz (p, f p ).
Proof.We let q 0 = p, f q 0 = f p and we construct q along with f q,s , s ∈ Split α (q α ), by a slow fusion sequence p α | α < λ .In step α the function f pα,s will be defined for s ∈ Split α (p α ).
In successor steps, given p α we define for s ∈ Split α (p α+1 ) a successor set Succ p α+1 (s) ⊆ Succ pα (s) and a function f p α+1 ,s as follows: Let the range of f pα,s be enumerated increasingly as η i | 0 ≤ i < κ α .We define the lower part of p α+1 such that for each s ∈ Split α+1 (p α ) = Split α+1 (p α+1 ) we have The f p,s is not a misprint here: This is the only stage of the construction when we refer to direct successors of s.Moreover, we define p α+1 such that Succ p α+1 (s) = dom(f p α+1 ,s ) is the set of these ξ's and for i < κ α , we let f p α+1 ,s (ξ) = η i < η i+1 = f p,s (ξ).The function f p α+1 ,s is also one-to-one and increasing.We define the higher part of the condition p α+1 by letting p α+1 ↾ t = p α ↾ t for t ∈ Succ p α+1 (s), s ∈ Split α+1 (p α+1 ).
In the limit steps α ≤ λ, we first take the intersection q α = {p β | β < α} and invoke Lemma 4.2.For α < λ, we strengthen q α to p α as in the successor step using Split α (q α ) as a starting point.According to Lemma 4.3, p α | α < λ is a slow fusion sequence.
In the end we let q be the fusion of p α , α < λ, and for s ∈ Split α (p α ) we let f q,s = f pα,s .Definition 4.11.A subset A ⊆ L is called an L-antichain if for any two (p, f p ), (q, f q ) ∈ A, p and q are incompatible.Lemma 4.12.Suppose (p, Proof.Suppose not.Then we may take q ≤ p such that q is incompatible with any member of π 1 (A).By Lemma 4.10 there is (r, f r ) ≺ horiz (q, f p ↾ q).Now (r, f r ) violates the L-maximality of A. Now we prove Theorem 4.7.
Proof.In the Miller case we work with the dense set of conditions p that have a layered sequence Split α (p) | α < λ of splitting maximal antichains and no other splitting.These are called weakly splitting normal conditions in [9, Def.2.8].These conditions correspond to the T ′ 's and Split α (T Nodes in the maximal antichain Split α (T ′ ) have κ α immediate successors in the condition.
By induction on n < ω we choose a sequence maximal L-antichains We let A 0 = {(p, id)} where id is the naturally defined identity function.
We carry out the successor step: For each (q, f q ) ∈ A n we take a maximal L-antichain A (q,fq) n+1 ≺ horiz -below (q, f q ) and we let A n+1 = {A (q,fq) n+1 |(q, f q ) ∈ A n }.
Now the rest of the proof is as in Theorem 4.7.We use a matrix A n | n < ω such that for any p ∈ A n and any q ∈ A n+1 , q ≤ p implies that q ≺ stat p.If p n | n < ω is any descending sequence such that p n ∈ A n for all n < ω, then p n | n < ω has no lower bound q, because for such a q the number n dom(Stem(q)) (OSucc q (Stem(q))) would not exist.
5. Failures of Three-Parameter Distributivity 5.1.A Cofinal Function from ω to κ in the Pre-Perfect Case.Let P be any of the pre-perfect versions and let λ be uncountable.In the Miller case, we restrict the forcing to the dense set of conditions p in which only the nodes in {Split α (p) | α < λ} have more than one successor.
We use the technical notion of a minimal branch.
In the Laver case, p pre (dom(Stem(p pre ))+α) = Split pre (p pre , α).In addition, for any q ≤ p pre we have q = q pre .Definition 5.3.We call p standardized if p = p pre .The construction in Definition 5.2 implies the following: Fact 5.4.The set of standardized pre-perfect trees is an open dense subset of P pre .5.2.The Singular Perfect Case.Now we further develop our non-distributivity results from Section 4. Theorem 5.11.Let κ be a singular cardinal of cofinality λ > ω and let P be any of M κ , L κ , M stat κ , or L stat κ .Then P is not (ω, •, λ + )-distributive.Lemma 5.12.Let p ∈ P be a condition and suppose that A = {q ξ : ξ < λ} is a maximal antichain below p.Then for all t ∈ p, there is some s ∈ p ↾ t and some ξ < λ such that p ↾ s ≤ q ξ .
Proof.Fix such a p ∈ P and A.
Claim: Fix ζ < λ.For all t, there is some t ∈ p ↾ t such that one of the following holds: (i) ∀ξ < ζ, (p ↾ t) ∩ q ξ ⊆ t, (ii) ∃ξ < ζ, p ↾ t ≤ q ξ .Suppose that the claim is false.The negation of this statement is that there is some t ∈ p such that ∀t ∈ p ↾ t, we have both of the following: (i') ∃ξ < ζ, s ∈ p ↾ t, s > t and s ∈ (p ↾ t) ∩ q ξ , (ii') ∀ξ < ζ, ∃s ∈ p ↾ t, s > t and s ∈ (p ↾ t) \ q ξ .This implies that for all ξ < ζ, t ∈ (p ↾ t) ∩ q ξ , there is some η < ζ, η = ξ and some s ∈ (p ↾ t) ∩ q η ; this is because we can first take s ∈ (p ↾ t) \ q ξ using (ii') and then s ′ ∈ (p ↾ s) ∩ q η for some η < ζ using (i'), from which it will follow that η = ξ.It then follows that we can find a ⊑-increasing sequence of nodes s i | i < ζ ⊂ p such that for all i < ζ, s ∈ q ξ(i) for ξ(i) < ζ and yet s i / ∈ q ξ(j) for j < i.If s = i<ζ s i , then s ∈ p. Then for all t ⊒ s with t ∈ p ↾ s and for all ξ < ζ, we have t / ∈ q ξ .Hence the claim holds as witnessed by s via (i).
Now suppose that the statement of the lemma is false: ∃ t ∈ p, ∀t ∈ p ↾ t, ξ < κ, p ↾ t ≤ q ξ .Suppose that P is one of the Miller versions.It follows from the claim that for all t ∈ p ↾ t, ζ < λ, there is some s ∈ p ↾ t such that ∀ξ < ζ, (p ↾ s) ∩ q ξ ⊇ t.We can therefore build a sequence A ζ | ζ < λ of maximal antichains in p ↾ t such that the following hold: (1) t ∈ A ζ , t is a splitting node, (2) for all ξ < ζ, (p ↾ t) ∩ q ξ ⊆ t, Then let r be the downwards closure of ζ<κ A ζ .Then r is a condition in P both in the Miller and Laver cases.Moreover, for all ξ < λ, r ⊥ q ξ , and so A is not maximal after all.
Here is the proof of Theorem 5.11.
Proof.Let {A n |n ∈ ω} be a matrix that witnesses the failure of ω-distributivity.Fix p ∈ P and suppose for contradiction that for all n < ω, |{r ∈ A n : r p}| ≤ λ.
so we conclude that forcing with either scl M club κ or L club κ induces a generic on Add(λ, 1).

Lemma 3 . 14 .
Suppose we are given p ∈ L κ .If b = b(p), then given any b ∈ Add(ω 1 , 1) such that b ≤ b, there is some p ≤ p such that b(p) = b.Proof.Fix p, b, and b.Let α be the length of the stem s of p.Let δ = dom(b).

4. 2 .
Miller Forcing and the Vertical Balcar-Simon Trick.Let P be M κ , M stat κ , M club κ .The following does not work either for the Laver case or for the splitting-closed Miller versions.

Definition 1.14.
We enumerate the versions we consider below.(i)We write L = L stat if for any splitting node t ∈ p ∈ P, we have OSucc p (t) is a stationary subset of {β ∈ κ dom(t) | cf(β) = ω}.

1.16. The forcing notions scl M club
and Magidor.Previous arguments, like the ones in [22, Ch.XI, Section 3, especially 3.3., 3.4 and 3.6] and [6, Theorem 2.3], used an intermediate step in which regular trees are sought (see Theorem 2.2 in [6]).A tree p is called regular if there is a set {ℓ n | n < ω} ⊆ ω, such that for any n, any t [1,.Laver Forcing and the Horizontal Balcar-Simon Trick.Let P be M κ , scl M κ or L κ .Here the (< cf(κ))-closed versions are of course excluded.The version with stationary ordinal successor sets is also excluded and will be handled in the next subsection.If P is one of M κ , scl M κ , or L κ , then P is not countably distributive.Rather than changing splitting nodes to non-splitting nodes, we thin out successor sets.In this way, we prove Theorem 4.7, adapting Balcar and Simon's descending technique of their Theorem[1, Theorem 5.16].A pair (p, f p ) is called a labeled tree if it has the following properties: