On automorphisms of Boolean algebras embedded in P

150 (1996)

On automorphisms of Boolean algebras embedded in P (ω)/fin

by

Magdalena G r z e c h (Warszawa)

Abstract. We prove that, under CH, for each Boolean algebra A of cardinality at most

the continuum there is an embedding of A into P (ω)/fin such that each automorphism of A can be extended to an automorphism of P (ω)/fin. We also describe a model of ZFC + MA(σ-linked) in which the continuum is arbitrarily large and the above assertion holds true.

It is well known that, under CH (the continuum hypothesis), each Boolean algebra of cardinality at most 2

can be embedded in P (ω)/fin (see e.g. [5]). This implication cannot be reversed: there is a model of set theory in which 2

> ω

and the above conclusion still holds ([1]). It is also known that CH is equivalent to the following condition: each Paroviˇcenko algebra (i.e. algebra of cardinality 2

, atomless and having neither countable limits nor countable unfilled gaps) is isomorphic to P (ω)/fin. We begin by proving the following.

Proposition 1. If CH holds, then for every Boolean algebra A of car- dinality at most the continuum there is an embedding i : A → P (ω)/fin such that each automorphism of i(A) can be extended to an automorphism of P (ω)/fin.

P r o o f. Assume CH. Let A be a Boolean algebra of cardinality at most the continuum. We will construct an extension A

of A such that:

1. A

is a Paroviˇcenko algebra;

2. S

α<ω1

A

= A

, where (A

: α < ω

) is an increasing sequence of algebras satisfying the following two conditions:

(?) card A

≤ 2

, A

= A, A

= S

α<λ

A

for every limit λ < ω

,

1991 Mathematics Subject Classification: Primary 03E50; Secondary 03E05, 06E05.

This paper is the author’s PhD thesis presented to the Institute of Mathematics, Polish Academy of Sciences in November 1995.

[127]

(??) for every automorphism φ

of A

there exists an automorphism φ

of A

such that φ

⊆ φ

.

It is clear that if an algebra A

satisfies the above conditions then each automorphism of A can be extended to an automorphism of A

. Thus to prove our theorem it suffices to construct A

.

Fix a pairing k : ω

×ω

→ ω

(one-to-one and onto) such that k(ζ, ξ) ≥ ζ for all ζ, ξ < ω

. It remains to describe the successor step from α to α + 1.

Suppose that we have defined a sequence (A

: γ ≤ α) satisfying (inductive) conditions (?) and (??).

Let E

(φ, a, c) abbreviate the statement: φ is an automorphism of A

such that φ(a) = c.

Assume that at each stage γ ≤ α we chose an enumeration (x

: ξ < ω

) of the collection of the following families:

{(c

, d

: i, j < ω) : ∃φ ∀i, j < ω [E

(φ, a

, c

) ∧ E

(φ, b

, d

)]},

where (a

, b

: i, j < ω) is a countable ordered gap a

< a

< . . . < b

< b

of elements of A

, {(b

: i < ω) : ∃φ ∀i < ω E

(φ, a

, b

)}, where (a

: i < ω) is a decreasing chain of elements of A

, and the set of all atoms of the algebra A

. (Since we assumed CH, we have at most ω

objects to enumerate.)

We identify A

with the field B(X

) of open-closed subsets of the associa- ted Stone space X

. The ordinal α determines a certain object, namely x

, where ξ and ζ are ordinals such that k(ζ, ξ) = α (ζ < α). If x

is a family of chains or gaps, we take A

to be the subfield of P (X

) generated by A

= B(X

) and by

n b = \

i<ω

b

: (a

, b

: i, j < ω) ∈ x

o

when x

is a collection of gaps, or by n

b = \

i<ω

b

: (b

: i < ω) ∈ x

o

when x

consists of countable chains. Using the Sikorski theorem (on exten- ding homomorphisms, see e.g. [7], [5]) we extend each automorphism of A

to an automorphism of A

and therefore (?), (??) hold.

Now, suppose that x

is a set of nonzero elements of A

. Each element

of the family is an atom of A

but it need not remain an atom in A

. If

there are at least countably many elements e

< a, a ∈ x

, then we put

A

= A

. (Note that the property (??) implies that if some element of

x

is an atom then all elements of the set are atoms.) Suppose that each

a ∈ x

is a finite sum of atoms a = e

+ . . . + e

. (n is the same for all

elements of x

by (??).) Atoms of A

correspond to isolated points of X

. We delete the isolated points e

< a for all a ∈ x

, and put into their places copies of a one-point compactification of the discrete ω. Let X

denote the topological space thus obtained. We put A

= B(X

). Since X

is a continuous image of X

, we have A

⊆ A

. Obviously, each automorphism of A

can be extended to an automorphism of A

. This finishes the proof.

Now we consider the case of ¬CH. It is known that there exists a model of ZFC+MA+¬CH in which the algebra P (ω

) is not embeddable in P (ω)/fin ([2]). On the other hand, it is consistent with ZFC and MA(σ-linked) that the cardinality of the continuum is arbitrarily large and each Boolean algebra of cardinality ≤ 2

can be embedded in P (ω)/fin ([1]). Thus the existence of such embeddings does not imply CH. The assertion of Proposition 1 is stronger and we may ask if the converse holds. The answer is negative. We prove that:

Theorem 1. It is consistent with ZFC + MA(σ-linked) that the cardina- lity of the continuum is arbitrarily large and for each Boolean algebra B of cardinality ≤ 2

, there is an embedding i : B → P (ω)/fin such that each automorphism of i(B) can be extended to an automorphism of P (ω)/fin.

P r o o f. Let V be a ground model satisfying the generalized continuum hypothesis (GCH). Thus there exists a regular cardinal κ in V such that κ > ω

and if κ = λ

, then cf(λ) > ω, moreover ♦

(the diamond principle) holds in the form:

There is a sequence (T

: α < κ, cf(α) = ω

) such that for every set X ⊆ H(κ) the set {α < κ : cf(α) = ω

, X ∩ H

= T

} is stationary in κ.

H(κ) denotes as usual the family of all sets of hereditary power < κ, H(κ) = S

α<κ

H

, and (H

: α < κ) is a continuously increasing sequence of sets of cardinality < κ.

We will define a finite support iteration (P

: α < κ) having the c.c.c.

(countable chain condition) such that, in the corresponding generic exten-

sion V[G], the conclusion of Theorem 1 will be satisfied. The model V will

be extended in such a way that given a system of generators for a certain

Boolean algebra B (card B < 2

) there will be an embedding sending the

generators to generic sets added at some steps α < κ. The embedding will

be defined by induction: If a certain monomorphism embeds the subalgebra

B

of B generated by the initial α (α < 2

) generators and if the image

(under the monomorphism) of each of them is a generic subset of P (ω)/fin

then the next generator determines in B

two sets (which form a gap): one

consists of elements less than the generator (called the “lower class”), and

the elements of the other (called the “upper class”) are disjoint from the generator. An image of the gap is a gap in the subalgebra of P (ω)/fin which is generated by some generic sets. The monomorphism can be extended if there is an element of P (ω)/fin which fills this gap. Thus, to ensure embed- dability of Boolean algebras, we will add generic sets X

⊆ ω filling gaps in subalgebras of P (ω)/fin generated by some previously added X

, β < α.

Simultaneously, in a similar way, we will extend automorphisms of the sub- algebras. In constructing embeddings of Boolean algebras and extensions of their automorphisms, we have to avoid the following problem: It is well known that there are gaps in P (ω)/fin which are unfillable by c.c.c. forcing.

It could happen, unless steps are taken to prevent it, that an image (under an extension of an automorphism of one of the embedded algebras) of some gap filled in a later step is an unfillable gap.

To ensure that every automorphism of an embedded Boolean algebra can be extended we use the ♦ principle. It guarantees that each such automor- phism is “approximated” by an increasing sequence of automorphisms which belong to models V[G|α]. To be more precise: if F is a canonical P

-name for some automorphism f of a given algebra B (from the model V[G]) then there is a subset A of κ such that

card A = κ and [

α∈A

(F ∩ H

) = [

α∈A

T

and T

is a P

-name for an automorphism from V[G|α]. We will extend automorphisms using those of the T

’s which are their names. To obtain MA(σ-linked) we will enumerate at some stages (with repetition) all σ-linked forcings R with card R < κ (cf. [1], [4]).

Assume the following notation:

Let X be a set and let T

denote a homomorphism. Then for ε ∈ {−1, 1}, εX denotes X, if ε = 1, or \X, if ε = −1. Moreover, T

is T

, if ε = 1, or T

, if ε = −1. (We abbreviate (T

)

to T

.)

For ϕ : α → {0, 1} let B(ϕ) be the subalgebra generated by {X

: ϕ(β) = 1}, where X

is a generic subset of ω added at stage β. If s is a finite sequence with dom(s) ⊆ {β : ϕ(β) = 1} and rg (s) ⊆ {−1, 1} then

X(s) = \

s(ξ)=1

X

∩ \

s(ζ)=−1

(ω \ X

).

Thus B(ϕ) consists of finite unions of sets of the form X(s). A gap in B(ϕ) is a system of the form

L = ({X(s) : s ∈ S}, {X(t) : t ∈ T }),

where X(s) ∩ X(t) =

∅ for all s ∈ S and t ∈ T .

An increasingly ordered gap L of type (λ, γ) is a gap as above such that there are enumerations S = {s

: α < λ} and T = {t

: β < γ} satisfying

α

< α

< λ ⇒ X(s

) ⊆

X(s

), β

< β

< γ ⇒ X(t

) ⊆

X(t

).

We assume that each gap except the increasingly ordered ones satisfies the condition: if s

, . . . , s

∈ S and X(s) ⊆

X(s

) ∪ . . . ∪ X(s

) then s ∈ S (and similarly for T ).

We will use two notions of forcing: Kunen’s forcing filling a gap, and the other, which adds an uncountable antichain to Kunen’s forcing of type (ω

, ω

).

Now we describe the two forcings:

Let L = ({X(s) : s ∈ S}, {X(t) : t ∈ T }) be a gap. Kunen’s forcing Q(L) consists of elements of the form (u

, x

, w

), where u

and w

are finite subsets of S and T (respectively) and x

is a finite zero-one sequence.

Moreover,

[

s∈u_q

X(s) ∩ [

t∈w_q

X(t) ⊆ dom(x

).

Let p = (u

, x

, w

) and q = (u

, x

, w

); then p is an extension of q (written p ≤ q) iff u

⊆ u

, w

⊆ w

, x

⊆ x

and for each i with dom(x

) ≤ i <

dom(x

), if i ∈ [

s∈u_q

X(s) then x

(i) = 1 and if i ∈ [

t∈w_q

X(t) then x

(i) = 0.

It is known that if L is separated, then Q(L) has the c.c.c.

Now let L = ({X(s

) : α < ω

}, {X(t

) : β < ω

}) be an increasingly ordered gap. A condition of forcing E(L) is a finite set e consisting of se- quences of the type (α, s

, t

) such that if (α, s

, t

), (β, s

, t

) ∈ e and α 6= β then either X(s

) ∩ X(t

) 6= ∅ or X(s

) ∩ X(t

) 6= ∅. E(L) is ordered by inverse inclusion. It is well known that if L is an unfilled gap then E(L) has the c.c.c. and

E(L) ° “Q(L) has an uncountable antichain”.

The definition of the iteration is inductive and uses a “bookkeeping”

technique. At each inductive step α < κ we enumerate some objects in V

, and at higher stages we add some generic sets to them. The objects occur in an order determined by a function Nb. To be more precise, we divide κ into five unbounded sets:

A = {α < κ : cf(α) = ω

}, M = {α ∈ κ \ A : α is odd},

E = k(A), Q

= k(M ), Q

= k(κ \ (A ∪ M )),

where k : κ → κ \ (A ∪ M ) is an increasing bijection.

Let {ν

: α < κ} be an increasing enumeration of the set {β < κ : β ≥ λ}, if κ = λ

, or of the set {β < κ : β is a cardinal and cf(β) > ω}, if κ is a limit cardinal.

Let n : κ × κ → κ be a pairing function satisfying:

ξ, ζ < n(ξ, ζ) for all ξ, ζ < κ, n(α, β) ∈ M for all α ∈ M, β < κ, n(α, β) ∈ Q

for all α ∈ Q

, β < κ, n(α, β

) < n(α, β

) for all α ∈ A, β

< β

< κ, n(α, β) ∈ Q

for all α ∈ A, β < ν

, n(α, β) ∈ E for all α ∈ A, β > ν

,

n(α

, β

) < n(α

, β) for β

< ν

, α

< α

, β < κ.

Using this function we will define (by induction) a function Nb. At stages ξ ∈ M , we will add generic filters to σ-linked forcings. In steps ξ ∈ Q

, we add (by Kunen’s forcing) the generic set X

which fills a gap consisting of some sets previously added (in Q

steps). In the model V[G], each Boolean algebra will be embedded in a certain algebra generated by sets obtained in these steps. At stage ξ ∈ Q

we also add (by the same forcing) the generic set X

which separates a gap, but this gap is generated by sets previously added both in Q

and Q

steps. In the model V[G] each of these sets X

, ξ ∈ Q

, will be an image (under one of the extended automorphisms) of some element of P (ω)/fin which appeared in some model V[G|δ], δ < ξ. In steps ξ ∈ E we will add uncountable antichains to Kunen’s forcing to keep gaps in the ranges (of the extended automorphisms) unfilled.

The sequence in which new elements of P (ω)/fin appear is important in our construction. It will be described by the function Ind from P (ω)/fin into κ, defined inductively simultaneously with iteration. We begin with the condition: if x ∈ P (ω)/fin ∩ V then Ind(x) = 0. At each higher stage we extend the function Ind according to the rule:

If P

° “x 6∈ dom(Ind) and x ∈ P (ω)/fin” then Ind(x) = α + 1.

If ξ < κ, cf(ξ) = ω

and P

° “T

is an automorphism of B(ϕ)” (T

is an element of the ♦-sequence), then we begin to define (inductively) families of monomorphisms according to the following conditions:

(a) T

= T

. (b) For γ ≥ ξ,

P

° “T

is a monomorphism from a subalgebra of P (ω)/fin into P (ω)/fin”.

(c) If γ

< γ

then T

is an extension of T

.

(d) If γ

≤ γ

, 0 < ξ

≤ γ

, P

° “T

is an automorphism of B(ϕ

)”

(i = 1, 2) and

P

° “For some ordinal %, ϕ

¹ % = ϕ

¹ % and

T

and T

agree on B(ϕ

¹ %)”, then

T

¹ {X ∈ P (ω)/fin : Ind(X) < %} ∩ dom(T

1

)

= T

2

¹ {X ∈ P (ω)/fin : Ind(X) < %} ∩ dom(T

2

).

(e) If λ ≤ α is a limit ordinal then T

= S

γ<λ

T

.

At each stage we will compute card P (ω)/fin using the following two (well known) theorems (see e.g. [5], [6]):

Theorem 2. Assume that P has the c.c.c. in V and let ν be a cardinal in V such that V ° “card P ≤ ν, ν

= ν”. Let Q be such that P ° “card Q ≤ ν”.

Then card P ? Q ≤ ν in V.

Theorem 3. Assume that P has the c.c.c. in V and λ, ν ≥ ω are cardi- nals in V such that V °“card P ≤ ν and λ = ν

”. Let G be P-generic over V. Then 2

≤ λ in V[G].

Thus we have to show that for each α, P

has the c.c.c. We will do that in the second part of the proof; now we assume that it is true.

We describe the inductive step α ⇒ α + 1. Assume that the forcing P

and families of monomorphisms T

(ξ ≤ γ ≤ α) satisfying the above conditions (a)–(e) are already defined. Assume also that card P

≤ ν

and P

° “2

≤ ν

”. Since the cardinality of each of the forcings occurring in Cases 1 to 5 below is ≤ ν

, by Theorems 1 and 2 we have card P

≤ ν

and P

° “2

≤ ν

”.

We distinguish five cases. In Cases 1, 2 and 5 we set T

= T

. C a s e 1: α ∈ M . We enumerate all P

-names of σ-linked forcings of cardinality < κ so that each forcing occurs κ times in the enumeration and the following holds:

If R is ξth element of the enumeration then P

° “card R ≤ ξ”.

We extend the function Nb: if

P

° “R is σ-linked and card R < β ”

and R is the βth element of the above enumeration then Nb(R) = n(α, β).

If there are γ < α and β < κ such that α = n(γ, β), then we put P

= P

? R,

where P

° “R is σ-linked and card R < κ” and Nb(R) = α.

C a s e 2: α ∈ Q

. In this case we enumerate all pairs (L, ϕ) of P

-names such that P

forces the following properties:

(a) ϕ ∈ D

, where D

consists of all ψ ∈ V

with dom(ψ) ≤ α, rg(ψ) ⊆ {0, 1} and Γ = {γ : ψ(γ) = 1} ⊆ Q

, and such that if {γ

: ξ < δ}

is an increasing enumeration of Γ , then for each ξ < δ there is a gap L

in B(ψ ¹ γ

+ 1) satisfying γ

= Nb(L

, ψ ¹ γ

+ 1).

(b) L is a gap in B(ϕ).

Each of these pairs occurs κ times in the enumeration.

If there are γ < α and β < κ such that n(γ, β) = α then we put P

= P

? Q(L),

where P

° “L is a gap in B(ϕ)” for some ϕ ∈ D

. C a s e 3: cf(α) = ω

. If

P

° “For some γ < α and ϕ ∈ D

, T

is an automorphism of B(ϕ)”

then two cases are possible:

(?) P

° “There is no ξ < α such that T

is an automorphism of B(ψ) and T

and T

agree on B(ψ

), where ψ

= ϕ ¹ %

= ψ ¹ %

for some ordinal

%

≤ ξ”, and

(??) P

° “There are ordinals ξ, %

and a function ψ ∈ D

such that

%

≤ ξ < α, cf(ξ) = ω

, ψ

= ϕ ¹ %

= ψ ¹ %

, T

is an automorphism of B(ψ) and T

is an extension of T

¹ B(ψ

)”.

Let Υ denote the set of all pairs (%

, ξ) such that P

forces that T

is an automorphism of B(ψ), ψ

= ϕ ¹ %

= ψ ¹ %

and T

is an extension of T

¹ B(ψ

). Let ζ = sup{%

: (%

, ξ) ∈ Υ }. We enumerate all triples (X, T

, εn) of P

-names such that

P

° “X ∈ P (ω)/fin”,

and Ind(X) < dom(ϕ) (case (?)), or ζ ≤ Ind(X) < dom(ϕ) (case (??)), ε ∈ {−1, 1}, n ∈ ω. We fix a function j from the set of these triples into κ with the following properties:

(a) If X ∈ B

= {X

: γ ∈ Q

, γ < dom(ϕ)} and Y 6∈ B

then

j((X, T

, εn)) < j((Y, T

, εm)) for all n, m < ω.

(b) If X

, X

∈ B

[resp. Y

, Y

6∈ B

] and Ind(X

) < Ind(X

) [resp. Ind(Y

) < Ind(Y

)] then

j((X

, T

, εn)) < j((X

, T

, εm)) [resp. j((Y

, T

, εn)) < j((Y

, T

, εm))].

(c) For all X with Ind(X) < dom(ϕ) and all n ∈ ω,

j((X, T

, −n)) = j((X, T

, n)) + 1,

j((X, T

, n + 1)) = j((X, T

, −n)) + 1.

Since P

° “card P (ω)/fin ≤ ν

”, the domain of the sequence of the triples is ≤ ν

.

Using ordinals > ν

we also enumerate all triples (L, T

, εn) of P

-names such that P

forces: L = ({X(s

) : γ < ω

}, {X(t

) : β < ω

}) is an increasingly ordered gap of the type (ω

, ω

), Ind(X(s

)) ≤ dom(ϕ) and Ind(Y (t

)) ≤ dom(ϕ) for all γ < ω

and β < ω

, and Q(L) does not have the c.c.c. We can assume that (L, T

, −n) follows (L, T

, n) and precedes (L, T

, n + 1).

We extend the function Nb to the set of objects described above in the following way:

Nb((X, T

, εn)) = n(α, j(X, T

, εn))

and if (L, T

, n) is the βth element of the (second) enumeration then Nb((L, T

, n)) = n(α, β).

We set

P

= P

.

We also define T

= T

= T

in case (?) and T

= T

= monomor- phism generated by T

and S

(%ξ,ξ)∈Υ

T

¹ {X ∈ P (ω)/fin : Ind(X) < %

} in case (??). The families T

defined at earlier stages are not changed:

T

= T

.

It is easy to check by using Sikorski’s theorem and the following lemma that the above definitions are correct.

Lemma 1. Let X be an element of P (ω)/fin in V, let p = (u

, x

, w

) ∈ Q(L) and let X

stand for a generic subset added by Q. Then we have:

if p ° “X ⊆

X

” then X ⊆

[

s∈up

X(s),

if p ° “X ∩ X

=

∅” then X ⊆

[

t∈wp

X(t).

C a s e 4: α ∈ Q

. Suppose that α = Nb((X, T

, εn)), where P

° “T

is an automorphism of B(ϕ), X 6∈ B(ϕ)

.

If ε = 1 and X 6∈ dom(T

) or ε = −1 and X 6∈ rg(T

) then we extend the monomorphism T

.

Suppose that ε = 1. Let L be a gap in rg(T

) defined by X:

L = ({(T

)

(Z) : Z ⊆

X}, {(T

)

(Y ) : X ⊆

Y })

(P

forces all the properties). All elements of the gap have been defined at the previous stages, because of the definition of j (Case 3). We set

P

= P

? Q(L).

We extend T

setting

T

= homomorphism generated by T

∪ {((T

)

(X), X

)}, where

X

= {i ∈ ω : ∃p ∈ G [x

(i) = 1]}, and G ⊆ Q(L) is a generic filter. If T

is a P

-name such that P

° “T

is an automorphism of B(ψ) and

for some ordinal %, ϕ ¹ % = ψ ¹ % and T

and T

agree on B(ϕ ¹ %)”, and Ind(X) < dom(B(ψ)), then

T

= homomorphism generated by T

∪ {((T

)

(X), X

)}, and T

= T

in the remaining cases. If ε = −1 we proceed with the construction in a similar way: we add a generic set to the domain of T

and to the domains of each of the T

’s which agree with T

on an “initial segment” of their domains.

(It is easy to prove, by using Lemma 1 and Sikorski’s theorem, that the definitions of the monomorphism T

are correct.)

C a s e 5: α ∈ E. Assume that α = Nb((L, T

, εn)), where P

° “L is an increasingly ordered gap in P (ω)/fin and Q(L) does not have the c.c.c.”

Suppose that L = ({X(s

) : ζ < ω

}, {X(t

) : β < ω

}) and let L

denote the gap

({(T

)

(X(s

)) : ζ < ω

}, {(T

)

(X(t

)) : β < ω

}).

We set

P

= P

? E(L

) and T

= T

.

For limit ordinals λ < κ we define P

as a direct limit of {P

: α < λ}.

We also assume P

= P

in all cases not mentioned above. This completes the definition of the iteration.

We conclude this part of the proof by checking that the above construc- tion is correct, i.e. that there is no gap L of the type (ω

, ω

), consisting of generic subsets of ω, which is an image (under one of the extending monomorphisms) of some gap L

such that Q(L

) does not have the c.c.c.

and which is filled by the generic set X

at some stage γ < κ.

Claim 1. Let B

(i = 1, 2) denote one of the following subalgebras of P (ω)/fin : B(ϕ

) (where ϕ

∈ D

); the domain of T

; the range of T

. Assume that S

i=1

X(s

) ∈ B

, S

j=1

X(t

) ∈ B

and S

i=1

X(s

) ⊆

S

j=1

X(t

). Then there are finite functions r

, . . . , r

such that rg(r

) ⊆ {−1, 1} (l = 1, . . . , k) and for each ξ ∈ S

l=1

dom(r

) we have X

∈ B

∩ B

and

[

X(s

) ⊆

[

X(r

) ⊆

[

X(t

).

This is proved by using Lemma 1.

Lemma 2. Let α < κ. Assume that P

has the c.c.c. Suppose that P

forces the following:

(1) For each β < α and each ϕ ∈ D

, if L

is a gap in B(ϕ), then Q(L

) has the c.c.c.

(2) If L

= (S

, U

) is a gap in the domain or range of (T

)

such that P

° “ Q(L) has the c.c.c.” and for all X(s) ∈ S

and X(t) ∈ U

there are S

i=1

X(s

) ∈ S

and S

j=1

X(t

) ∈ U

such that X(s) =

[

X(s

) ∧ X(t) = [

X(t

)

∧ Ind

 [

i=1

X(s

)



< Ind

 (T

)

 [

i=1

X(s

)



∧ Ind

 [

j=1

X(t

)



< Ind

 (T

)

 [

j=1

X(t

)



then P

° “Q((T

)

(L)) has the c.c.c.”

(3) L

= ({(T

)

(X(s

)) : X(s

) ∈ S}, {(T

)

(X(t

)) : X(t

) ∈ U}) is an increasingly ordered gap such that P

° “T

is an automorphism of B(ψ), L = (S, U) is an increasingly ordered gap of the type (ω

, ω

) in P (ω)/fin and Q(L) does not have the c.c.c.”

Under the above assumptions, there is an α

< ω

such that for each γ > α

and any finite subsets {X(s

), . . . , X(s

)}, {X(t

), . . . , X(t

)} of the lower and upper classes (respectively) of the gap L

or L

the following holds:

X(s

) 6⊆

[

X(s

) or X(t

) 6⊆

[

X(t

)

where X(s

), X(s

) and X(t

), X(t

) are elements of the lower and upper classes of L

(respectively).

P r o o f. Assume to the contrary that for every γ ∈ ω

there are s

, . . . . . . , s

, t

, . . . , t

such that

h

X(s

) ⊆

[

X(s

) i

∧ h

X(t

) ⊆

[

X(t

) i

.

Applying Claim 1 to X(s

) ⊆

S

i=1

X(s

) and X(t

) ⊆

S

j=1

X(t

) (for each γ < ω

) we obtain elements S

i=1

X(r

), S

j=1

X(p

) such that X(s

) ⊆

S

i=1

X(r

) ⊆

S

i=1

X(s

), X(t

) ⊆

S

j=1

X(p

) ⊆

S

j=1

X(t

) and X

∈ B ∩ rg((T

)

) for each η ∈ S

i=1

dom(r

) ∪ S

j=1

dom(p

). (Here B denotes B(ϕ) or the domain or range of T

.) Thus

L

=

n [

i=1

X(r

) : γ < ω

o

, n

[

j=1

X(p

) : γ < ω

o

is a gap in B ∩ rg(T

).

If B = B(ϕ) then L

is a gap in B(ϕ ∩ ψ). Thus L

, the image of L

under (T

)

, is a gap in B(ψ) and by (1), Q(L

) has the c.c.c., but by (3) it does not have the c.c.c., a contradiction.

If B = dom(T

) then (T

)

(L

) = L

is a gap in dom((T

)

). By (2), Q(L

) has the c.c.c. but by (3) it does not have the c.c.c., a contradic- tion.

We show that the assertion of Theorem 1 holds in the extension V[G], where G ⊆ P

is a generic filter. It is clear that V[G] ° “2

= κ” and (by Theorems 2 and 3), V[G|α] ° “2

< κ” for each α < κ. Let B be a Boolean algebra in V[G] with card B = κ. There are elements b

∈ B for γ < κ such that B = S

α<κ

B

, where B

is the subalgebra generated by b

, γ ≤ α.

Assume inductively that we have an embedding i : B

→ P (ω)/fin such that i(b

) = X

with β

∈ Q

for each ξ < α. We define a sequence ϕ

: sup{β

: ξ < α} → {0, 1} putting ϕ

(β

) = 1 for each ξ < α, and ϕ

(ζ) = 0 otherwise. Thus B(ϕ

)/fin is an isomorphic image of the algebra B

. Let

b(s) =  Y

s(ζ)=1

b



·

 Y

s(η)=−1

−b

 ,

where s is a finite function on α with rg(s) ⊆ {−1, 1}. The next generator b

determines a gap

L

= ({b(s) : b(s) ≤ b

}, {b(t) : b(t) · b

= 0})

in the algebra B

. Let L be the image of L

under i. So L is a gap in B(ϕ

) and

L = ({X(s

)}, {X(t

)}),

where s

is defined on {β

: ξ ∈ dom(s)} by the equality s

(β

) = s(ξ) (t

is defined similarly).

Let γ > sup(ϕ

), γ ∈ Q

and γ = Nb(L, ϕ

). We define i(b

) = X

and

ϕ

= ϕ

∪ {(β, 0) : dom(ϕ

) < β < γ} ∪ {(γ, 1)}. This extends i to an

embedding from B

onto B(ϕ

)/fin (we check this using Lemma 1).

Let Φ = S

α<κ

ϕ

. It is clear that B is isomorphic to B(Φ).

Let f be an automorphism of B. Then i ◦ f ◦ i

is an automorphism of B(Φ) and there is a canonical name F for it, F ⊆ H(κ), consisting of some pairs ((x, y)

, p), where x, y are canonical names for the elements of B(Φ) and the set F (x, y) = {p ∈ P

: ((x, y), p) ∈ F } is an antichain. Since P

has the c.c.c., the set

N

= {α < κ : ∀x, y [x, y ∈ V

→ F (x, y) ⊆ P

]}

is ω

-club (closed and unbounded) in κ. For any α ∈ N

the restriction F

= F ∩ (V

× P

)

is a P

-name and F

[G|α] = i ◦ f ◦ i

∩ V[G|α]. So, for all α ∈ N

, the monomorphism F

[G|α] belongs to V[G|α]. On the other hand, the sets

N

= {α < κ : β < α, cf(α) = ω

, F ∩ H

= F

}

are ω

-club for all β < κ. From the diamond principle it follows that there is an increasing sequence {γ

∈ N

∩ N

: β < κ} such that F

= T

. Let A(F ) = S

β<κ

T

and f = A(F )[G]. Then f is an automorphism of P (ω)/fin and i ◦ f ◦ i

⊆ f .

It remains to show that P

has the c.c.c. for each α ≤ κ. Let P

consist of all p ∈ P

satisfying the following conditions:

1. For each γ ∈ supp(p) ∩ (Q

∪ Q

) there are u

(p), x

(p), w

(p) such that

p ¹ γ ° “p(γ) = (u

(p), x

(p), w

(p))”

and dom(s) ⊆ supp(p) for each s ∈ u

(p) ∪ w

(p). Moreover, for each γ ∈ supp(p) ∩ (Q

∪ Q

), the number dom(x

(p)) is constant (independent of γ).

We write l(p) for this value.

2. For each γ ∈ supp(p) ∩ E there are (α

, s

, t

), . . . , (α

, s

, t

) such that

p ¹ γ ° “p(γ) = {(α

, s

, t

), . . . , (α

, s

, t

)}”

and dom(s

) ∪ dom(t

) ⊆ supp(p) for i ≤ n.

Let P

⊆ P

be the set of all p ∈ P

with the property:

3. If γ ∈ M then there is an n ∈ ω such that p ¹ γ ° “h

(p(γ)) = n”, where h

is a P

-name of a function such that

P

° “h

: R

→ ω and ∀n ∈ ω [h

(n) is linked]”.

(We can choose the h

since P

° “R

is σ-linked”.)

Lemma 3. For each p ∈ P

and m ∈ ω, there is a q ∈ P

such that p ≥ q

and l(q) ≥ m.

P r o o f. The proof (except for the case β ∈ E) is similar to the proof of Lemma 4.4 in Chapter 9 of [5].

Assume (inductively) that the lemma holds for α, β = α+1, β ∈ supp(p) and β ∈ E. There is a p

≤ p¹ β such that

p

° “p(β) = {(α

, s

, t

), . . . , (α

, s

, t

)}”

for some P

-names (α

, s

, t

), . . . , (α

, s

, t

). We may assume that dom(s

) ∪ dom(t

) ⊆ supp(p) for i ≤ n. By the inductive assumption there is a p

≤ p

such that p

∈ P

and l(p

) ≥ m. Thus, the element p

? p(β) has all the required properties.

We precede the next two lemmas with the following note: Fix α < κ and suppose that P

has the c.c.c. and the assumptions of Lemma 2 are satisfied.

Let P

force that L is the image under T

of an increasingly ordered gap L

such that

P

° “Q(L

) does not have the c.c.c.”

Suppose that {p

: ξ < ω

} ⊆ P

is a set of pairwise compatible conditions and that e

are P

-names of conditions of the forcing E(L) such that

∀ξ < ω

[p

° “e

= {(α

, s

, t

1

), . . . , (α

, s

, t

nξ

)}”].

Let z

, i = 1, . . . , n, be finite functions with dom(z

) ⊆ T

ξ<ω₁

supp(p

) ∩ (Q

∪Q

). From Lemma 2 it follows that there are (at most) two possibilities:

1. There is an uncountable set B ⊆ ω

such that

∀ξ

, ξ

∈ B [r

° “e

= e

”], where r

≤ p

, p

.

2. Any set A ⊆ ω

satisfying the following condition:

If ξ

, ξ

∈ A then for some i

∈ {1, . . . , n

} and j

∈ {1, . . . , n

} we have

p

° “X(s

) ⊆

[

X(z

)” and p

° “X(t

) ⊆

ω \ [

X(z

)”

and for all r

≤ p

, p

, r

° “∀k ∈ {1, . . . , n

} [α

0

6= α

] and ∀l ∈ {1, . . . , n

} [α

0

6= α

]”

is at most countable.

Lemma 4 ([5]). Let p ° “X(s) ∈ fin” and γ = max dom(s). If p ∈ P

then there is an r ∈ P

with r ≤ p and l(p) = l(r) such that if r ¹ γ ° “r(γ) =

(u

, x

, w

)”, then r ° “s ¹ γ ∈ u

” (if s(γ) = −1) or r ° “s ¹ γ ∈ w

” (if

s(γ) = 1).

Lemma 5. Assume that p, q ∈ P

satisfy the following conditions:

1. p¹ α and q ¹ α are compatible.

2. If ξ ∈ supp(p) ∩ supp(q) ∩ M and

p ¹ ξ ° “p(ξ) ∈ h

(n)” and q ¹ ξ ° “q(ξ) ∈ h

(m)”

then n = m.

3. If ξ ∈ supp(p) ∩ supp(q) ∩ (Q

∪ Q

) and

p ¹ ξ ° “p(ξ) = (u

, x

, w

)” and q ¹ ξ ° “q(ξ) = (u

, x

, w

)”

then x

= x

.

4. Let ξ ∈ supp(p) ∩ supp(q) ∩ E and p ¹ ξ ° “p(ξ) = {(α

, s

1

, t

1

), . . . , (α

, s

, t

nξ

)}”, q ¹ ξ ° “q(ξ) = {(β

, s

1

, t

1

), . . . , (β

, s

, t

mξ

)}”.

Define A

= {i : (α

, s

, t

i

) ∈ p(ξ) and α

6= β

for all j such that (β

, s

j

, t

j

) ∈ q(ξ)} (B

is defined in a similar way). Assume that for any i ∈ A

and j ∈ B

there is no s

with dom(s

) ⊆ supp(p) ∩ supp(q) such that

p ° “X(s

i

) ⊆

[

X(s

)” and q ° “X(t

j

) ⊆

ω \ [

X(s

)”.

Then p and q are compatible.

P r o o f. Denote by ∆ the set supp(p) ∩ supp(q) ∩ E. The required condi- tion will be constructed in the following way: First we define extensions of the conditions p and q by extending zero-one sequences x

and x

such that

p ¹ ξ ° “p(ξ) = (u

, x

, w

)” and q ¹ ξ ° “q(ξ) = (u

, x

, w

)”.

This will be done in such a way that if % ∈ ∆ and if some extension r of the conditions p and q forces

X(s

) = ε

X

1

∩ . . . ∩ ε

X

ni

, i ∈ A

, and

X(t

) = ε

X

1

∩ . . . ∩ ε

X

mj

, j ∈ B

, then for some n ≥ l(p),

x

k

(n) =  0, ε

k

= −1, 1, ε

k

= 1, x

l

(n) =

 0, ε

= −1, 1, ε

l

= 1, x

k

⊂ x

k

, x

l

⊂ x

.

Thus we obtain extensions p

and q

which force “n ∈ X(s

)” and “n ∈

X(t

)” respectively. In the next step of the proof we will consider the