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TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 174, December 1972

ADEQUATEULTRAFILTERS OF SPECIAL BOOLEANALGEBRAS BY

S. NEGREPONTIS(l) ABSTRACT. In his paper Good ideals

in fields

of sets Keisler

proved, with

the aid of the generalized continuum hypothesis, the existence of countably incomplete, /jf^-good ultrafilters on the field of all subsets of a set of (infinite) cardinality ß. Subsequently, Kunen has proved the existence of such ultrafilters, without any special set theoretic assumptions, by making use of the existence of certain families of large oscillation. In the present paper we succeed in carrying over the original arguments of Keisler to certain fields of sets associated with the homogeneous-universal (and more generally with the special) Boolean algebras. More specifically, we prove the existence of countably incomplete, ogood ultrafilters on certain powers of the ohomogeneous-universal Boolean algebras of cardinality cl and on the a-completions of the ohomogeneous-universal Boolean algebras of cardinality a, where a= rc-^ > w. We then develop a method that allows us to deal with the special Boolean algebras of cardinality ct= 2""". Thus, we prove the

existence of an ultrafilter p (which will be called adequate) on certain powers S * of the special Boolean algebra S of cardinality a, and the ex-

istence

of a specializing

chain

fC«:

ß < a\ for

oa,

such

that

C snp

is

/3+-

good and countably incomplete for ß < a. The corresponding result on the existence of adequate ultrafilters on certain completions of the special Boolean algebras is more technical. These results do not use any part of the generalized continuum hypothesis.

Keisler, simple

sets

in the proof of his fundamental

set-theoretic

of a given

results

set.

of his paper

of the "disjoint Indeed (Lemma

lemma on the "disjoint

In Question to arbitrary

refinement"

the results 2.1) proved

7 of his paper Boolean

lemma

of the ptesent for the Stone

result

space

he asks

algebras,

as a major

paper

stated

refinement"

above,

families

nature

a generalization.

of such

a lemma

of the homogeneous-universal

by the editors

of the

the special

to such

rely on the analogue

Presented to the Society, May 22, 1970; received and, in revised form, July 5, 1972.

of sub-

for generalizations

considering

obstacle

made use of a

of certain

September

Boolean

19, 1970

AMS (MOS)subject classifications (1970). Primary 02J05, 02K25, 06A40, 54H10; Secondary 02H13, 08A05, 12L10, 54G10. Key words and phrases. Powers and completions of special, homogeneous-universal Boolean algebras, topological characterization of Stone space of a-homogeneous-universal Boolean algebras, existence of adequate, good, countably incomplete ultrafilters, degree of goodness, weakly ^atomic Boolean algebras, generalized continuum hypothesis. (*) The author acknowledges partial support received from the Canadian National Research Council under grant A-4035. The results of this paper were announced in the

Notices

Amer. Math. Soc. on May 1970 [n,]. Copyright © 1973. American Mathematical Society

345 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

346

S. NEGREPONTIS

algebras.

However,

characteristic in 1.4(b)

the proof of Lemma

properties

below),

characteristic

Lemma

The paper

[K

An almost

a-complete

by L (4.2).

geneous-universal"

In §2,

to the proof of Theorem

on the existence

with the

of Pierce

(described the second

in 1.4(c) below)

In Part I we consider

of cardinality

a-completion

a-generated

algebras

of Part I use

(given

albegras.

of homogeneous-universal

a (Theorem

be identified

Boolean

of adequate

algebras

completions

cardinality

in two parts.

leading

the proof is given

filters

algebras

2.2, which is the analogue

are established,

Boolean

Boolean

the results

Boolean

and special

Part I on the existence special

use only of one of the two

2.2).

geneous-universal

refinements

2.1 makes

of homogeneous-universal

while

property

(in proving

[December

of special

in §3.

We are

connected

as one would wish;

cardinals

with also

and for the strong

limit cardinals. The results extreme)

cases

of both Parts of a general

in the remarks

[Ko],

of Part

to those

Galvin

[G],

of the generalized

cumbersome ization

II may be used

given

by Keisler

as particular

completions,

and especially continuum

for various

[Kj],

which

equivalence

themselves

applications.

1. Preliminaries.

deeper

The axiom of choice

by ¿5 £, 77, À with or without

of ultrapowers results

without

(indeed, is outlined

[K-r],

making

Kochen

any use are rather

[Sh] on the characterthey will not be given. I do not seem to lend

Ordinal

An ordinal

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

applications,

however

of Part

is assumed.

subscripts.

[Kg],

applications

work of Shelah

in terms

and in a sense

to similar

[Mj], [M9], These

and in view of the definitive

of elementary

model-theoretic

[K9], [Kg], [Kr],

Mansfield

hypothesis.

The more satisfactory

denoted

on partial

at the end of §4.

The results analogous

I and II may be considered

procedure

numbers

coincides

are

with the

1972]

ADEQUATE ULTRAFILTERS OF SPECIAL BOOLEAN ALGEBRAS

set of all smaller

ordinals,

make the notational A cardinal

i.e.

distinction

number

is an initial

0, 1, • • • , 72, k, « • • denote least

cardinal

mappings

£ < ( is equivalent

greater

between

the first

ordinal.

Cardinals

natural

numbers.

than ß is denoted

from ß to cl is denoted

(G.C.H)

states

that

cl

= 2

whenever

be identified

with

empty subsets

For a nonempty

the cofinality If a. is equal

cardinal,

otherwise

a whenever

ß>

ß,

denote

limit

A is de-

the set of all nonX (X 4 0),

if a = cf(a),

singular

cl is called

a nonlimit

then

satisfying

the condition

2^
of cardinality (2) Boolean

from 1.1(a),

cofinality,

limit cardinal;

inaccessible;

holds for all (infinite)

as the generalized

The general

algebras.(2)

or a. is a strong

and only if a is regular

or cl ¡s (strongly)

(f) [G.C.H.]

tions,

cl = a^

cf(A);

less

Alternatively, algebras (cf.

has been developed

is the text of Bell-Slomson

[BS].

only as they apply to the jónsson

algebra

than

classes

C is a-homogeneous

cl, and any embeddings

we may consider Lemma 3 in [JO]).

the complete

in [Jj],

[Jol» [Ko]»

We need here the class

of Boolean

if for any Boolean

algebra

h., by J> ~' C, there is an and model-complete


theory

of atomless

348

S. NEGREPONTIS

automorphism

if given

h oí C, such that

any Boolean

algebra

h °h.

If C is a-homogeneous

universal.

For the notion of special,

alent to the original definition a be an (infinite)

nality

|l»:

ous-universal 1.2. cardinality gebra

at most

2 . Then,

C of cardinality

statement

of C,

where

there

case

than

a.

We need

and let

is a ß

C.

Co is

the following

facts. algebra

of C

by Morley-Vaught;

of cardito the union

ß -homogene-

be any Boolean

embedding

(equiv-

[CK].

special

C is equal

-homogeneous-universal algebra

of a result

definition

Ç is called

subalgebras

2 , and a Boolean

is a particular

algebra

i.e.

cardinal

a-homogeneous-

simplified

chain,

ß less

ß be any infinite

a-universal

is an embedding

then C is called

a specializing

of Boolean

C is

there

IMV]), given by Chang-Keisler

A Boolean

for all cardinals Let

a,

we adopt the following

by Morley-Vaught

cardinal.

ß < a\

algebra

at most

and a-universal,

a if \C\ - a and if C has

of a chain

A Boolean

J^ of cardinality

h: S)—'C

Let

= h2.

[December

of

Boolean

al-

into

(This

C.

cf. Theorem

2.10 and

Remark (4) following Theorem 2.8 in [MV]). 1.3. algebra more,

Let

a = 2A

of cardinality

a.

if a is regular,

universal

Boolean

There

(This

then

algebra

is up to isomorphism Boolean

algebra

&a is the unique, of cardinality

exactly

a.

one special

will be denoted

by aa.)

up to isomorphism, (This

statement

Boolean Further-

a-homogeneous-

is a particular

case

and compact

zero-

of Theorem 2.8 in [MV].) We make

use

dimensional 5(C).

of Stone's

spaces.

We identify

field of sets space

and

such that

The Stone

of an appropriate

U is equal

between

space

C with the field

U an open

the following 1.4.

duality

of a Boolean

of S.

of S(t). The type

Let

C be an (infinite)

SiL)

has no isolated

(b)

if U is open and 1 < r(u)
be a field

the notion definition

set

only for the

is a sequence

introduce

Let

a

S be a zero-dimensional

subsets

cardinal

of S.

a,

We need

then

W, such that in Corollary

U is not dense

such that (/ C W and

algebra.

Then

assumption

\Z : n | of elements of a good ultrafilter A map

f on i

cp: So(ß)

< a,

then

1.7 in [N,];

they

algebra

of cardi-

for their proof.) is countably

of p such that

on a field

+ riv)

V n W - 0.

Boolean

is not needed

An ultrafilter

in S((?);

riu)

1.5 and Theorem

a-homogeneous-universal

of sets.

in (KJ.

Boolean

elements;

are given

a, but the additional Let

or with

conditions:

is an open-and-closed

nality

by

of S(C),

AU) of U is the least

a-homogeneous-universal

(a)

are stated

C is denoted

subsets

to the union of a open-and-closed

the following

(These

algebra

facts:

S(C ) satisfies

there

algebras

of open-and-closed

subspace

subspace

Boolean

of sets

11

Z

in analogy

—>A is monotone


incomplete

= 0.

if

We

to Keisler's

if cp(F) C cp(F

)

197 2]

ADEQUATE ULTRAFILTERS OF SPECIAL BOOLEAN ALGEBRAS

349

for F, F' £ SJ\ß), such that F A F', and multiplicative if (j>ÍFU F ' ) = 0(F) ft :SJ/3)

p on J

—>p, there

is a-good

if for every

is a multiplicative

ß < a and

mapping

ifi:

Sj.ß) -* p, such that 0(F) C (F)for ail F e SJ.ß) (denoted by xfj< ).We set F* = 3\!0|.

If E C F

the set of all monotone

and a is an infinite mappings

set of all multiplicative

we let Mon (a,

c/>: S J.ß) —»E fot all

mappings

S —* § be an embedding

cardinal,

ß < a,

E) denote

and Mult (a,

É) the

:$ J-ß) —' F fot all ß < cl. Let further

of fields

of sets.

There

are obvious

induced

h:

embeddings,

denoted by Mon (a, h): Mon (a, E) —>Mon (a, h[E]), etc.

An ultrafilter

Mon (a,

f on y is called

E), there

a general

Let

of an adequate

in each

duality,

algebra

the power

Si(¿) x 8, the set

8 being

S, we let

JoiS) denote

of S.

Thus,

and inMC)

C

if/ < Mon ia,

ultrafilter,

and let

C

given

space

called

if h[E] C p and for every

but refer

with the field

B and D for

the discrete

S. Let

topology. algebra

We identify, subsets

of

For a zero-dimensional

of all open-and-closed

x 8) ate isomorphic.

An ultrafilter

subsets

p on C

will be

e £ p, the set

h: C

epimorphism

cardinal.

of all open-and-closed

\X £

case.

C be a Boolean

via Stone's

E)-good

is tfj £ Mult (a, p) such that

definition

its meaning

(a,

» C,

e/0\

be a Boolean

of Stone spaces

algebra

is denoted

embedding.

by Sib):

5(C)

The induced —►SiC),

and

we let Sib) x ids: SicV/>.

77, such that

the conditions:

for ff c = ijr

53y C3)s for y