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It is shown that if various cardinal invariants of the continuum related to d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {d}}$$\end{document} are equal to ℵ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\aleph _1$$\end{document} then there is a non-trivial automorphism of P(N)/[N]<ℵ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {P}}({\mathbb {N}})/[{\mathbb {N}}]^{<\aleph _0}$$\end{document}. Some of these results extend to automorphisms of P(κ)/[κ]<κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {P}}(\kappa )/[\kappa ]^{<\kappa }$$\end{document} if κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} is inaccessible.


Introduction
A fundamental result in the study of theČech-Stone compactification, due to Rudin [7,8], is that, assuming the Continuum Hypothesis, there are 2 c autohomeomorphisms of βN\N and, hence, there are some that are non-trivial in the sense that they are not induced by any one-to-one function on N. While Rudin established his result by showing that for any two P-points of weight ℵ 1 there is an autohomeomorphism sending one to the other, Parovičenko [6] showed that non-trivial autohomeomorphisms could be found by exploiting the countable saturation of the Boolean algebra of clopen subsets of βN\N-this is isomorphic to the algebra P(N)/[N] <ℵ 0 . Indeed, the duality between Stone spaces of Boolean algebras and algebras of regular open sets shows that the existence of non-trivial autohomeomorphisms of βN\N is equivalent to the existence of non-trivial isomorphisms of the Boolean algebra P(N)/[N] <ℵ 0 to itself. Notation 1.1 If A and B are subsets of κ let ≡ κ denote the equivalence relation defined by A ≡ κ B if and only if |A B| < κ and A ⊆ κ B will denote the assertion that |A \ B| < κ. Let [A] κ denote the equivalence class of A modulo ≡ κ and let P(κ)/[κ] <κ denote the quotient algebra of the P(κ) modulo the congruence relation ≡ κ . If κ = ω it is customary to use ≡ * instead of ≡ ω and ⊆ * instead of ⊆ ω .

Notation 1.2
If f is a function defined on the set A and X ⊆ A then the notation f (X ) will be used to denote { f (x) : x ∈ X } in spite of the potential for ambiguity. Definition 1.3 An isomorphism : P(κ)/[κ] <κ → P(κ)/[κ] <κ will be said to be trivial if there is a one-to-one function ϕ : κ → κ such that ([A] κ ) = [ϕ(A)] κ for each A ⊆ κ. The isomorphism will be said to be somewhere trivial if there is some B ∈ [κ] κ and a one-to-one function ϕ : B → κ such that ([A] κ ) = [ϕ(A)] κ for each A ⊆ B and will be said to be nowhere trivial if it is not somewhere trivial.
The question of whether the Continuum Hypothesis, or some other hypothesis, is needed in order to find a non-trivial isomorphism of P(N)/[N] <ℵ 0 to itself was settled in the affirmative by Shelah in [9]. The argument of [9] relies on an iterated oracle chain condition forcing to obtain a model where 2 ℵ 0 = ℵ 2 and every isomorphism of P(N)/[N] <ℵ 0 to itself is induced by a one-to-one function from N to N. The oracle chain condition requires the addition of cofinally many Cohen reals and so d = ℵ 2 in this model. Subsequent work has shown that it is also possible to obtain that every isomorphism of P(N)/[N] <ℵ 0 is trivial by other approaches [2,10,14] but these have always required d > ℵ 1 as well. However, it was shown in [11] that this cardinal inequality is not entailed by the non existence of nowhere trivial isomorphisms from P(N)/[N] <ℵ 0 to itself-in the model obtained by iterating ω 2 times Sacks reals there are no nowhere trivial isomorphisms yet d = ℵ 1 .
On the other hand, while we now know that the Continuum Hypothesis cannot be completely eliminated from Rudin's result, perhaps it can be weakened to some other cardinal equality such as d = ℵ 1 . It will be shown in this article that non-trivial isomorphisms of P(N)/[N] <ℵ 0 to itself can indeed be constructed from hypotheses on cardinal arithmetic weaker than 2 ℵ 0 = ℵ 1 and reminiscent of d = ℵ 1 . However, it is shown in [3] that it is consistent with set theory that d = ℵ 1 yet all isomorphisms of P(N)/[N] <ℵ 0 are trivial so some modification of the equality d = ℵ 1 will be required.
It will also be shown that natural generalizations of the arguments can be applied to the same question for P(κ)/[κ] <κ where κ is inaccessible. The chief interest here is that, unlike P(N)/[N] <ℵ 0 , the algebra P(κ)/[κ] <κ is not countably saturated if κ > ω-to see this, simply consider a family {A n } n∈ω ⊆ [κ] κ such that n∈ω A n = ∅. In other words, Parovičenko's transfinite induction argument to construct non-trivial isomorphisms from P(κ)/[κ] <κ to itself is not available and some other technique is needed.
The statement and proof of Lemma 2.1 is provided for all κ and we will apply both to the case that κ = ω and to the case that κ is inaccessible. However, Lemma 3.2 deals only with the case that κ is inaccessible. It is somewhat simpler than the case for ω and so is dealt with first because the general approach is similar in the κ = ω case, but this case requires some technical details not needed in the inaccessible case.

A sufficient condition for a non-trivial isomorphism
The following lemma provides sufficient conditions for the existence of a non-trivial isomorphism of P(κ)/[κ] <κ to itself. The set theoretic requirements for the satisfaction of these conditions will be examined later. The basic idea of the lemma is that an isomorphism of P(κ)/[κ] <κ can be approximated by partitioning κ into small sets I ν and constructing isomorphisms from subalgebras of P(I ν ) and taking the union of these. Unless the subalgebras of P(I ν ) are all of P(I ν ), this union will only be a partial isomorphism. Hence a κ + length sequence of ever larger families of subalgebras of P(I ν ) is needed to obtain a full isomorphism. In order to guarantee that this isomorphism is not trivial the prediction principle described in hypothesis (H 3 ) of the lemma is needed.
and begin by observing that this is well defined. To see this, it must first be observed that given A and B such that |A B| < κ there is α ∈ κ + such that for all ξ > α and for all ν in a final segment of κ the equation is defined and valid by hypothesis (H 4 ). From hypothesis (H 2 ) it then follows that if ξ and η are greater than α then and, hence, ([A] κ ) is well defined. Since each ξ,ν is an automorphism it follows that is an automorphism of P(κ)/[κ] <κ . To see that is non-trivial, suppose that there is a one-to-one function F : It follows from hypothesis (H 2 ) that for any η ≥ ξ

When are the hypotheses of Lemma 2.1 satisfied?
In answering a question of A. Blass concerning the classification of cardinal invariants of the continuum based on the Borel hierarchy Goldstern and Shelah introduced a family of cardinal invariants called c( f, g) defined to be the least number of uniform trees with g-splitting needed to cover a uniform tree with f -splitting [4] and showed that uncountably many of these can be distinct simultaneously. The following definition is very closely related to this as well as to the notion of a slalom found in [1].
Lemma 3.2 Let κ be inaccessible and f : κ → κ and g : κ → κ be functions such that: • both f and g take their values in the set of cardinals, then the hypotheses of Lemma 2.1 hold.
Proof Given the hypothesis, it may be assumed that there are ⊆ κ -descending sets (This is done simply by reindexing so that for all ξ ∈ κ + there are cofinally many η ∈ κ + such that G ξ = G η .) Moreover, by a diagonal argument using the fact that lim ν→κ g(ν) = κ it can be assumed that if ξ ∈ η then G ξ (ν) ⊆ G η (ν) for a final segment of ν ∈ X η . (This is the part of the argument that does not extend to the case be a bijection. Let A 0,ν and A 1,ν partition I ν into two sets of cardinality f (ν) and let ϕ 0,ν be an involution of I ν interchanging A 0,ν and A 1,ν . For ν ∈ X 0 let B 0,ν = {∅, I ν , A 0,ν , A 1,ν } and let 0,ν be the automorphism of B 0,ν induced by ϕ 0,ν . For ν ∈ κ \ X 0 let B 0,ν = P(I ν ) and let 0,ν be the identity. As the induction hypothesis assume condition (H 2 ) of Lemma 2.1 holds and that, in addition, If B ξ,ν , A ξ,ν , ϕ ξ,ν and ξ,ν have been defined for all ξ less than the limit ordinal η then a standard diagonalization yields B η,ν , A η,ν , ϕ η,ν and η,ν .
To see that hypothesis (H 3 ) of Lemma 2.1 holds let F : κ → κ be one-to-one. If there are cofinally many ν such that there is z ν ∈ I ν such that F(z ν ) / ∈ I ν then let ξ = 0 and, without loss of generality, it may be assumed that z ν belongs to the atom A 0,ν of B 0,ν for cofinally many ν.
If, on the other hand, F(I ν ) ⊆ I ν for all ν in a final segment of κ then F I ν = θ J (ν),ν for some J (ν) also for a final segment. There is then some ξ ∈ κ + such that J (ν) ∈ G ξ (ν) for all ν in a final segment of X ξ and it may as well be assumed that ξ is a successor. By construction, for all ν in a final segment of X ξ there is Finally, to see that hypothesis (

The special case of κ = ω
The proof of Lemma 3.2 does not apply to κ = ω because it relies on the fact that μ·μ = μ if μ is an infinite cardinal. This is used to reduce to the case G ξ (ν) ⊆ G η (ν) for most ν if ξ ∈ η. The first part of the proof of Lemma 4.1 corrects this-the function f in its hypothesis can be thought of as yielding a sequence of integers approximating infinite cardinals-but the general outline of the proof is the same as that of Lemma 3.2. Proof Given the hypothesis, it may be assumed that there are ⊆ * -descending sets {X ξ } ξ ∈ω 1 ⊆ F and functions {G ξ } ξ ∈ω 1 ⊆ n∈ω [ f (n)!] g(n) such that for every F ∈ n∈ω f (n)! there is ξ ∈ ω 1 such that F(n) ∈ G ξ (n) for all but finitely many n ∈ X ξ . (This is done simply by reindexing so that for all ξ ∈ ω 1 there are cofinally many η ∈ ω 1 such that G ξ = G η .) It will first be shown that it can be assumed that there are functions, g, h ξ : N → N and H ξ ∈ n∈N [ f (n)!] h ξ (n) for ξ ∈ ω 1 such that (a) lim n→∞ f (n)/g(n)2 g(n) = ∞, (b) if ξ ∈ η ∈ ω 1 then 4h ξ ≤ * h η ≤ g, (c) if ξ ∈ η ∈ ω 1 then H ξ (n) ⊆ H η (n) for all but finitely many n, (d) if F ∈ n∈N f (n)! and F(n) ∈ G ξ (n) for all but finitely many n ∈ X ξ then also F(n) ∈ H ξ (n) for all but finitely many n ∈ X ξ .
To see this note that the hypothesis that lim n→∞ f (n)/g(n)k g(n) = ∞ for all k makes it possible to choose h : N → N such that lim n→∞ f (n) g(n)h(n)2 g(n)h(n) = ∞ and lim n→∞ h(n) = ∞. Then find h ξ : N → N for ξ ∈ ω 1 such that if ξ ∈ η then 4h ξ (n) < h η (n) < h(n) for all but finitely many n ∈ N. Let g(n) = h(n)g(n) and note that it can be assumed that f (n) > g(n)2 g(n) for all n ∈ N. Define h ξ (n) = g(n)h ξ (n) and observe that if ξ ∈ η ∈ ω 1 then for all but finitely many n.
Let H 0 (n) = G 0 (n). Given H ξ satisfying conditions (c) and (d), define H ξ +1 (n) = H ξ (n) ∪ G ξ +1 (n) and note that On the other hand, if η is a limit ordinal and H ξ satisfying the desired requirements have been chosen for ξ ∈ η, then a diagonalization argument yields H η such that |H η (n)| = h η (n) and H ξ (n) ⊆ H η (n) for all but finitely many n for each ξ ∈ η. Hence h ξ and H ξ satisfy conditions (b), (c) and (d).
Now let {I n } n∈ω partition N so that |I n | = f (n) and let {θ j,n } j∈ f (n)! enumerate all permutations of I n . Let A n : f (n)! → P(I n ) be a surjection. Without loss of generality, f (n) is even for each n. So let A 0,n and A 1,n partition I n into two equal sized sets and let ϕ 0,n be an involution of I n interchanging A 0,n and A 1,n . For n ∈ X 0 let B 0,n = {∅, I n , A 0,n , A 1,n } and let 0,n be the automorphism of B 0,n induced by ϕ 0,n . For n ∈ ω \ X 0 let B 0,n = P(I n ) and let 0,n be the identity. As the induction hypothesis assume condition (H 3 ) of Lemma 2.1 holds and that, in addition, (e) A ξ,n are the atoms of B ξ,n and that |A ξ,n | ≤ 2 4h ξ (n) provided that n ∈ X ξ , (f) for n ∈ X ξ there are involutions ϕ ξ,n of I n that induce ξ,n .
If B ξ,n , A ξ,n , ϕ ξ,n and ξ,n have been defined for all ξ less than the limit ordinal η then a standard diagonalization yields B η,n , A η,n , ϕ ξ,n and η,n .
To see that hypothesis (H 4 ) of Lemma 2.1 holds let F : N → N be one-to-one. If there are infinitely many n such that there is z n ∈ I n such that F(z n ) / ∈ I n then let ξ = 0 and, without loss of generality, it may be assumed that z n belongs to the atom A 0,n of B 0,n for infinitely many n. Since ϕ 0,n (A 0,n ) = A 1,n ⊆ I n it is clear that F(z n ) / ∈ ϕ 0,n (A 0,n ). If, on the other hand, F(I n ) ⊆ I n for all but finitely many n then F I n = θ J (n),n for some J (n) also for all but finitely many n. By condition (d) there is some ξ ∈ ω 1 such that J (n) ∈ H ξ (n) for all but finitely many n ∈ X ξ and condition (c) allows the assumption that ξ is a successor. By construction, for all but finitely many n ∈ X ξ there is a singleton {k} ∈ A ξ,n such that ξ,n ({k}) = {ϕ ξ,n (k)} and ϕ ξ,n (k) = θ J (n),n (k) = F(k).
Finally, to see that hypothesis (H 5 ) of Lemma 2.1 holds let A ⊆ N. Let F ∈ n∈N f (n)! be such that A∩ I n = A n (F(n)) for all n. By the hypothesis of Lemma 3.2 there is ξ ∈ ω 1 such that F(n) ∈ G ξ (n) for all n ∈ X ξ . From condition (d) it follows that F(n) ∈ H ξ (n) for all but finitely many n ∈ X ξ . Since B ξ,n = P(I n ) if n / ∈ X ξ it follows that A ∩ I n ∈ B ξ,n for all but finitely many n ∈ N. Proof In this case let F be the co-finite filter and note it is generated by the constant ⊆ * -descending sequence all of whose terms are ω.

Corollary 4.3 If there is an
Proof Let F be generated by {X ξ } ξ ∈ω 1 . Use Rothberger's argument and ℵ 1 = d to construct a ⊆ * -descending sequence {Y ξ } ξ ∈ω 1 all of whose terms are F positive and such that Y ξ ⊆ X ξ . Let F be generated by {Y ξ } ξ ∈ω 1 and note that d f !,g (F ) = ℵ 1 .
It has to be noted that the hypothesis of Corollary 4.3 is not vacuous in the sense that there are models of set theory in which it holds. For example, in the model obtained by iterating Miller reals ω 2 times the following hold: • d = ℵ 2 because the Miller reals themselves are unbounded by the ground model, • d f,g = ℵ 1 for appropriate f and g because the Miller partial order satisfies the Laver property, • u = ℵ 1 because P-points from the ground model generate ultrafilters in the extension.
However there does not seem to be any model demonstrating that the assumption that ℵ 1 = d in Corollary 4.3 is essential. It is shown in [3] that it is consistent with set theory that d = ℵ 1 yet all automorphisms of P(N)/[N] <ℵ 0 are trivial. However, u = ℵ 2 in that model because random reals are added cofinally often.

Remarks and questions
The first thing to note is that there are models where d f !,g = ℵ 1 < 2 ℵ 0 for f and g satisfying the hypotheses of Lemma 4.1-for example, this is true in the model obtained by either iteratively adding ω 2 Sacks reals 1 or adding any number of Sacks reals, greater than ℵ 1 of course, side-by-side. Of course d = ℵ 1 also in these models. It is therefore of interest to note that the Laver property implies that d f !,g = ℵ 1 in the Laver model as well, yet d = ℵ 2 in this model. It should also be observed that it is possible for d f,g to be larger that d. For example, iteratively forcing ω 2 times with perfect trees T that are cofinally f branching will yield such a model. To be a bit more precise, given f : ω → ω define S( f ) to consist of all trees T ⊆ n∈ω j∈n f ( j) such that or each t ∈ T there is s ⊇ t such that s j ∈ T for all j ∈ f (|s|). So Sacks forcing is just S(2) where 2 is the constant 2 function. The same proof as for Sacks forcing shows that S( f ) is proper and adds no reals unbounded by the ground model. Iterating S( f ) with countable support ω 2 times then yields model in which d = ℵ 1 . However, if g : ω → ω and H ⊆ n∈ω [ f (n)] g(n) has cardinality ℵ 1 then there is some model containing g and H and there is ∈ n∈ω f (n) which is generic over this model. This generically ensures that for all h ∈ H there is some j such that ( j) / ∈ h( j). The generalization of Sacks reals to uncountable cardinals in [5] establishes that the hypotheses of Lemma 3.2 can be satisfied for uncountable cardinals. Alternatively, one could iteratively force κ + times with the partial order P( f, g) consisting of pairs (G, F) where |G| < κ and G is a function whose domain is a subset of κ and G(ξ ) ∈ [ f (ξ )] g(ξ ) and F ⊆ ξ ∈κ f (ξ ) and |F| < κ with the ordering defined by (G, F) It is easy to see that P( f, g) is κ-closed and that if ⊆ P( f, g) is generic then the domain of (G,F)∈ G has cardinality κ and the family of such domains added iteratively will generate the necessary filter.
However, the following question does not seem to be answered.
It is worth observing that the isomorphism of Lemma 2.1 is trivial on some infinite sets-indeed, if ξ ∈ κ + and X ⊆ N are such that {x} belongs to some B ξ,ν for each x ∈ X then is trivial on P(X ). However, if T( ) is defined to be the ideal {X ⊆ N : P(X ) is trivial} then T( ) is a small ideal in the sense that the quotient algebra P(N)/T( ) has large antichains, even modulo the ideal of finite sets-in the terminology of [2], the ideal T( ) is not ccc by fin. To see this, simply observe that the proof of Lemma 4.1 actually shows that hypothesis (H 4 ) of Lemma 2.1 can be strengthened to: For any one-to-one F : N → N there is ξ ∈ κ + such that for all but finitely many ν ∈ ω there is an atom a ∈ B ξ,ν and ι ∈ a such that F(ι) / ∈ ξ,ν (a). It follows that if Z ⊆ N is infinite then Z * = ν∈Z I ν / ∈ T( ). Hence, if A is an almost disjoint family of subsets of N then {A * : A ∈ A} is an antichain modulo the ideal of finite sets.
One should not, therefore, expect to get a nowhere trivial isomorphism by these methods. It is nevertheless, conceivable that there are some other cardinal invariants similar to d f,g that would, when small, imply the existence of nowhere trivial isomorhisms of P(N)/[N] <ℵ 0 . In this context it is interesting to note that it is at least consistent with small d that there are nowhere trivial isomorphisms. Sketch of proof The partial order defined in [12, Section 2, Definition 2.1] will be used 2 . Begin with a model V satisfying 2 ℵ 0 > ℵ 1 and construct a tower of permutations {(A ξ , F ξ , B ξ )} ξ ∈Lim(ω 1 ) such that, letting S η = {(A ξ , F ξ , B ξ )} ξ ∈Lim(η) and P η be the finite support iteration of partial orders that are Q(S ξ ) for ξ ∈ Lim(η) and Hechler forcing if ξ is a successor, the following holds for each η and G that is P ω 1 generic over V : The proof of [12,Theorem 2.1] shows that there is a nowhere trivial isomorphism of P(N)/[N] <ℵ 0 in this model and, since P ω 1 is ccc, it is also true that 2 ℵ 0 remains larger than ℵ 1 in the generic extension. The Hechler reals guarantee that d = ℵ 1 .
It should also be noted that Lemma 2.1 actually yields 2 (κ + ) isomorphisms. It is shown in [13] that it is possible to have non-trivial isomorphisms of P(N)/[N] <ℵ 0 without having 2 c such isomorphisms. This motivates the following, somewhat vague, question.

Question 5.3
Can there be some variant of d f,g which, when small, yields a nontrivial isomorphism of P(N)/[N] <ℵ 0 without yielding the maximal possible number of such?
Given the remarks following Corollary 4.3 it is natural to ask the following. As a final remark it will be noted that Corollary 4.2 shows that [12, Theorem 3.1] cannot be improved to show that in models obtained by iterating Sacks or Silver reals all isomorphisms of P(N)/[N] <ℵ 0 are trivial because the equality d f !,g = ℵ 1 holds in these models for the necessary f and g.