DISTRIBUTIVITY AND THE NORMAL COMPLETION OF BOOLEAN

DISTRIBUTIVITY

AND THE NORMAL COMPLETION

BOOLEAN ALGEBRAS

R. S. PIERCE

1. Introduction. In a recent paper, [4], Smith and Tarski studied

the

interrelations between completeness and distributivity properties of

Boolean algebra. Independently, the author also obtained some of the

results of Smith and Tarski. This work was reported in [2]. The present

paper

continues the study of distributivity in Boolean algebras. Specifi-

cally, it deals with the problem of imbedding a Boolean algebra B in

α-distributive, /^-complete algebra, a and β being infinite cardinal

numbers.

If it is required that the imbedding be regular, that is, preserve

existing joins and meets, then (see [3]) the problem is equivalent to the

question of when the normal completion of B (or a subalgebra of the

completion)

is α-distributive. Our two main results can be briefly stated

follows

THEOREM

3.1. Every

a-distributive

Boolean

algebra

can be

regularly

imbedded

in an

a-complete,

a-distributive

Boolean

algebra.

THEOREM

5.1. There

exists

a-field

sets

whose

normal com-

pletion

is not

a-distributive.

Between these principal results, we obtain two simple conditions,

one

of which is necessary, the other sufficient for the normal completion

of a Boolean algebra to be α-distributive. These appear naturally as

particular

cases of more general facts relating properties which are

similar to, but not identical with α-distributivity and ^-completeness.

2. Preliminary results. The notation of this paper

will

be the same

as that of [2]. The Greek letters α, β and γ

always

denote cardinal

numbers,

while p,

and τ are used as indices belonging to sets R, S

and

T respectively. The symbol oo

will

be used as though it were

largest cardinal. This is a notational convenience, and in no case in-

volves

questionable logic. As in [2], a subset A of an arbitrary Boolean

algebra B is called a

covering

(of B) if the least upper bound of A in

B is the unit u of B. If the elements of the covering A are disjoint,

then

A is termed a

partition.

Finally, if the covering ^partition) A is

of cardinality

less

than,

or equal to a, symbolically A^α, then A is

called an α-covering (respectively, α-partition). If A and A are sub-

sets of B, then A is said to refine A when every aeA is<£some aeA.

Received

December

28, 1956. In revised form July 11, 1957.

133

134 R. S.

PIERCE

DEFINITION

2.1 (Smith-Tarski). A Boolean algebra B is called (a,

β)

-distributive if

ΛV

\/Λbί)

F=T

holds

identically when S^a, T^β and the bounds are assumed to exist

Some

elementary consequences of this definition are worth noting :

(2.2) If B is (α, /^-distributive and a'-^a, β

^β

then B is (a\ β')~

distributive. Any regular

subalgebra of an (a, β)-distributive Boolean

algebra is (a, β)-distributive. Every Boolean algebra is (n, β)-distributive,

where n is finite and β is arbitrary.

The

last assertion of (2.2) is a variant of the Tarski-von Neumann

theorem

(see [1], p. 165). This infinite distributivity is a property of

Boolean

algebras which we use repeatedly and without mention.

useful characterization of (α, /^-distributive Boolean algebras is

given by the following theorem, which, in somewhat different terms,

appears

in [4]. Since this characterization is used often in the sequal,

we sketch a proof.

THEOREM

2.3. Let a and β be

arbitrary

cardinal numbers. A Boolean

algebra

B is (a, β)-distributive if and only if, for any family {A

\σ e S} of β-

coverings

of B

with

S^a

there

is a

covering

of B

which

refines

every

Proof

Suppose B is (^/^-distributive. Let

{AJσ-eS}

be a given

family of /5-coverings with S^a. It can be assumed that every A

indexed by the same set T: A

={a

στ

\τ^T}. Let A~{aeB\{a} refines

every A

). Clearly A refines every A

. If A is not a covering of B, there

exists bφO (the zero of B) which is disjoint from every as A. Setting

στ

Λb, it is easy to see that AaVA

=&>0= VψΛAψoo

Thίs con

tradicts

(α, β)-distributivity. Thus A is a covering.

Conversely, let B satisfy the condition of the theorem. Suppose

Vrerbσr,

esVτ6ϊ*crr=& and Λ

€

A<^(σ) exist for all

G S

Siπd

all ψ e F

. Let ω be a symbol not in T. Put T = T[j{ω}, 6^=6', A

={b

στ

τeT'},b

ΛaeΦσφ(σ)

for all φeF. Then each A

is a /9-covering, so by

assumption

there is a covering A which refines every Ά

. If a e A, then

either

a<^b

for some φeF, or else a<^b\ Thus, if

c^>b

for all

φ>

cVδ'^l.u.b. A^u (the unit of B). Hence, c^b. Since b is obviously

upper bound of all b

ψ1

it follows that b=/\

φeF

For

simplicity, an (a, α)-distributive B. A. is just called α-distribu-

tive.

A subalgebra B of a Boolean algebra B is called regular (see [3]) if, whenever a

—

l.u.b. A in B (a£B, A^B),

then

α=l.u.b.

A in B also. Of course, in a Boolean al-

gebra,

this property implies its dual and conversely.

DISTRIBUTIVITY

AND THE NORMAL COMPLETION OF BOOLEAN 135

COROLLARY

2.4. A

Boolean

algebra

B is

a-distributive

if and

only

every

family {A

\σ e S) of

binary

partitions

with

S^La has a

common

refining

covering.

Indeed,

\<r

e S} (S^a) is a family of α-coverings, say A

στ

\τeT}

then, setting

στ

=[α

στ

(O'], the set

στ

\σeS,

reT} is

family of no more than a binary partitions of B and any covering which

refines all A

σr

is a common refinement of all A

(because

στ

—

For

future reference, we

list

some of the

well

known properties of

the

normal completion (or completion by " cuts ") of a Boolean algebra.

The

Stone-Glivenko theorem ((2.5) below) is proved in the standard re-

ference [1], The proofs of (2.6) to (2.8) are conveniently collected

[3].

(2.5) (Stone-Glivenko) The normal completion of a Boolean algebra

is a Boolean algebra.

(2.6) Let B be the normal completion of the Boolean algebra B.

Then

B is a regular subalgebra of B.

(2.7) Any Boolean algebra B is dense in its normal completion B.

That

is, if

OΦbeB,

then there

exists

beB with k

(2.8) If the Boolean algebra B is a dense subset of the complete

Boolean algebra B, then B is isomorphic to the normal completion of

Moreover, if BaBaB and B is complete, then B—B.

DEFINITION

2.9. Let B be a Boolean algebra. Let B be the normal

completion

of B. Let a be an infinite cardinal number. The

normal

completion

of B is the intersection of all α-cornplete subalgebras of B

which contain B. Denote this algebra B

. It

will

also be convenient

write B°° for B.

Clearly, B

is the smallest α:-complete subalgebra of B°° containing

Moreover, B is dense in B* and is regularly imbedded in B*.

3. The imbedding theorem. The primary purpose of this section

is to prove Theorem 3.1 (stated in the introduction). However, the

method

of the proof is used several times in the following sections, so

it behooves us to present it in a form which is sufficiently general to

cover all future needs.

LEMMA

3.2. Let B be a

complete

Boolean

algebra.

Let %_be a non-

empty

family of

partition

of B

such

that if {A

\σ e S} gSί and S^α, then

some

A e 2ί

refines

every

. Let B be the set of all

joins

subsets

the

partitions

A in Si. Then B is an

ct-complete

Boolean

algebra

such

136 R. S.

PIERCE

that A^B for

every

A 6 Si and

every

a-covering

of B is

refined

some

Ae3ί. Hence, B is

a-distribittive.

Proof

If CgAe 31, then (l.u.b. C)' = Lu.b. (A-C), since A is

partition. Hence B is closed under complementation. Suppose {c

\σ e S}

is a subset of B with S<^a. By definition of B, for each σeS, there

exists a partition A^eSI and a subset

such that c

=l.u.b. C

Then

refines the binary partition {c

, (c

)'}. Let A e 2ί be a common

refinement

of all A

. Then A is a common refinement of all {c

, (c

)'}

and

g.l.b. {c

|σ-eS}=l.u.b.

{aeA\a<Zc

all o eSjeB. Indeed, c = l.u.b.

{αeA|α<^c

, all σeSJ^v^σ is clear. But also, c' = l.u.b. {αeA|α<;

(Co)', seme <reS}^Λ

σe

ιJίCσ)'

=(Λ

eίPo)'- Hence, B is an α-complete B.A.

Obviously, A^B for all A 6 St. If A is an α:-covering of B, then, as

proved above, every binary partition {c, c'} with c e A is refined by some

e St. Choosing A e §ί to be a refinement of all these A

gives

a refine-

ment

of A. In fact, any aeA satisfies either α<^c, or a<Lc' for all

eel If a^c' for every c, than α^Λ

/ =

(l.u.b. A)' = 0, since A is

covering. Thus every aeA satisfies a<^c for some eel

Proof

of (3.1). Let 5 be the normal completion of B. Let 3ί be

the

set of all partitions of B, which are of the form Π

€)S

= {b

\φe 2

where the A

~ {a

σ0

, α

σl

} are binary partitions of B and b

= Λ

σe

,$Ar<κo

)

6 J?.

The

fact that Π^e^Ao- is a partition follows directly from the assumed

distributivity of B. If A

σesω

σr

e^L for all reϊ

with T^a, then

A=Π

τer

σesCτ

Sί is a common refinement of all A

. Thus, the

hypotheses of (3.2) are satisfied. Consequently, there is an α:-complete,

α-distributive Boolean algebra B with Bξ^BξΞ^B. Since B is a regular

subalgebra of B, it is also a regular subalgebra of B.

4 Conditions for distributivity. In this section, we

will

examine

the

following

five

properties of a Boolean algebra B :

( I

) B is α-complete

( II

) every subset of an ^-partition of B has a l.u.b. in B

(IΠβ)

every /3-covering of B can be refined by a /^-partition

(7F

Λβ

)

B is (α, /^-distributive

( V

Λβ

) If

\o-eS}

is a set of /9-partitions of B with S^a, then there

is a covering of B which is a common refinement of every A

Certain

relations between these properties are more or

less

evident.

(4.1) (a) I

and II

are hereditary in a, that is, I,, implies I

and

II"

implies Π

for all γ<^a

(b)

Λβ

and V^β are hereditary in both a and /3

(c)

implies Π

(d)

Λβ

implies V

Λβ

DISTRIBUTIVITY

AND THE NORMAL COMPLETION

BOOLEAN

137

(e)

Λβ

and IΠ

together imply

Λβ

(f)

if I

holds

for all a<β,

then

ΠI

, is

satisfied

(g)

aΛ

equivalent

to V

and

hence

to V

ΛΛ

(h)

III"oo

always

satisfied,

so IV^ is

equivalent

to F^oo.

Proofs.

The

statements (a)-(e)

are

obvious.

α-complete

for

all

a<β, and

A—{a

}

is a

^-covering

of B

indexed

by the set of all

ordinals

ξ of

cardinality

less

than

β,

then

\ξ<β}

will

a/3-partition

refining

A if

Cξ=aξA(V

The

assertion

of (g) is a

restatement

(2.4).

Finally, with

the

help

Zorn's lemma,

it is

always

possible

construct

partition

refine

any

covering. This construction,

the de-

tails

which

omit, proves

(h).

appears from

(4.1)

(e)-(h) that

the

condition

Λβ

only

slightly

weaker than

aβ

the

other hand,

the

condition

substantially

weaker than

, as the

following

example indicates.

Let X be a set of

cardinality

β let B be the

Boolean algebra

finite subsets

of X and

their

complements.

If a is any

cardinal number

less

than

β,

then

any

α-partition

of B is

finite. Consequently,

satisfies

. In one

case

however,

the

properties

and II

are

equivalent, namely

(4.2)

//«, is

equivalent

Zoo.

Proof.

Let C be an

arbitrary subset

of B. Let

C'={deB\d*c

all

ceC}.

Then clearly,

u is the

only upper bound

of the set CuC,

that

is, CuC" is a

cover.

By (4.1)

(h), there

is a

partition

refining

CuC.

D={aeA\{a}

refines

C},

then

A-D= {ae

A|αΛC = 0,

all ceC}.

Hence

l.u.b.

C=l.u.b.

exists

by //«,.

appropriate

now to

explain

the

object

studying

the

various

properties listed above.

Our

main interest,

course,

is the

relation

between

/«, and

Λoύ

and

specifically

would

find simple neces-

sary

and

sufficient conditions

for the

normal completion

of a

Boolean

algebra

satisfy

ΛΛ

It is

rather

easy

prove that

aoo

sufficient

and

Λθ

(αθ

necessary

for

α-distributivity

Z?°°.

The

effort

to fit

these

two

facts into

broader pattern leads

consideration

conditions

and V

aβ

. It

turns

out

that properties

and V

aβ

are

tied together

rather

closely. Unfortunately

and

aβ

do not

enjoy such

intimate

relationship and

the two

conditions mentioned above

are the

less

accidental offspring

of Π

and V

aβ

rather than

the

progeny

of I

and

Λβ

THEOREM

4.3. // the

Boolean

algebra

satisfies

Λβ

and

where

y—β

then

satisfies

Proof.

The

theorem

trivial

if a is

finite,

so it

will

assumed

138 R. S. PIERCE

that

a is an infinite cardinal number. Let A be a ^-partition of B.

Then

A can be indexed by a subset of T

, where T=β and ~S=a, say

A={a

}. Since B

satisfies

, it is meaningful to define 6

στ

= l.u.b.

\φ(σ-)

= τ} for each σeS, τeT. Then A

={b

σr

\τeT} is a β-partition

of B and it is easy to see that any common refinement of all A

is also

refinement of A, Now suppose

\peR}

is a set of ^-partitions of

B and R<^a. For each p in β, define (as above) a set of /^-partitions

pσ

\σeS

}

with the property that a common refinement of every A

pσ

with o e S

is also a refinement of A

. Consider the set of all ^-parti-

tions

pσ

.\σ-eS

peR). There are at most a

-a of these, so by pro-

perty V

aβ

, there is a covering A which refines every A

pσ

. But then A

refines every A

. Thus, B

satisfies

COROLLARY

4.4 (Smith-Tarski [4]). IfB is

a-distributive

and ^-com-

plete,

then B is (a,

^-distributive.

COROLLARY

4.5. A

necessary

condition

that B

a-distributive,

where

β^2

, is that B be (a,

^-distributive.

Indeed,

if B

is ^-distributive, then by (4.4) it is (a, 2*)-distributive.

But B is a regular subalgebra of B

and hence (by (2.2)) B is also (a,

2")-distributive.

We do not know whether the converse of 4.5 holds. That is, if

B is (α, 2*)-distributive, does it

that B

%Λ

is α-distributive ? This

seems doubtful, but if the goal of 2*-completeness (that is, property

2<Λ

)

is replaced by the property II

ι0L

, then a positive result is obtained (in

Corollary 4.8 below).

THEOREM

4.6. Let B be an

arbitrary

Boolean

algebra.

Define

B to

be the

intersection

of all

algebras

with

the

property

such

that

B^BQB

. Then B

satisfies

Moreover,

B has

property

Λβ

if and

only

if B has

property

Λβ

Also,

if B is

a-complete

and

satisfies

aβ

where

—β, then B is

a-complete.

Proof.

Clearly B

satisfies

. Since B is a regular subalgebra of

By the property V

Λβ

for B implies the same property for B. To estab-

lish the converse, it is sufficient to show that every β-partition of B can

be refined by a β-partition of B.

Let SI be the set of all β-partitions of B. By (2.5), every A e Si can

be considered as a partition of B°°. By (2.2), every finite subset of 31

has a common refinement in SI. Let B be the set of all joins in B°° of

subset of some A e 31. By (3.2), B is a Boolean algebra containing B.

Clearly B^B. Suppose A is a /5-partition of B, say A— {a

\τ e T]. Then

DISTRIBUTIVITY

AND THE NORMAL COMPLETION OF BOOLEAN 139

=Vi&«rrkeSv} with

σr

eB,

b^b^^O

for σψσ\ and S

^β. Con-

sequently, A—

σr

\σ-

e S

, reT} is a /^-partition of B which refines A.

The

join of any subset of A is also the join of a subset of A and

therefore in B. Since A was an arbitrary /^-partition, B has property

//β.

Consequently, B^B. Thus every /9-partition of B=B can be re-

fined by a ^-partition of B.

Finally, suppose B is ^-complete and

satisfies

Λβ

, with β

~β. If

{ΛrKeS},

S^α: is a set of β-partitions of B, then ΠσesΛr={Λ

eA-l

6^4

}

is a β

=/3-partition. Hence, by (3.2),

Z?=J3

is α-complete.

COROLLARY 4.7. The normal

completion

of a

Boolean

algebra

B is

(<%,

)-distributive if and

only

if B is (a,

^-distributive.

Proof

By (4.6), (4.1) and (4.2).

COROLLARY 4.8. // the continuum

hypothesis

true

for the

infinite

cardinal a (that is, 2

covers

α), then an

a-complete

Boolean

algebra

can be

regularly

imbedded

in an

a-complete,

a-distributive

algebra

satis-

fying

zΛ

if and

only

if B is (a,

^-distributive.

Proof.

The sufficiency of (α, 2

)-distributivity is a consequence of

(4.6) and (4.1). The necessity

follows

from (4.3), (4.1) and (2.2).

5. An example. Because of (4.5), the Theorem (5.1) of the intro-

duction

can be proved by constructing an α-field which is not (α, 2*)-

distributive.

Let X be a set of cardinality 2*. Denote by Y the set of all ordinal

numbers

of cardinality

less

than a. Let Z be the set of all

bounded

func-

tions

in Y

, that is, functions / for which there is an η e Y such that

f[x)<η for all x in X. Let £? be the collection of all sets of the form

where WQX, W^a and ψ e Y

. It is obvious that -^contains the empty

set and is closed under α-intersections.

Let ^ be the a -field generated by ^. It is to be shown that S^

is not (a, 2*)-distributive. The proof hinges on a lemma, which is useful

its own right.

LEMMA 5.2. Let Z be a set.

Suppose

J5f is a nonempty family of

sub-

sets

of Z

with

the

following

properties

(i)

every

a-intersection

sets

in J2^ is in 5^;

(ii) the

complement

of any set of Jzf is a union of

sets

of JSf.

Let

ά^ be the

a-field

generated

by ~Sf. Then Jίf is

dense

140 R. S. PIERCE

Proof.

Let lδ be the complete B. A. of all subsets of Z. Let SI be

the

collection of all partitions A of 58 with

AgΞ-2^.

|σeS}£Ξ2l,

and say A

={L

σr

\τeT

}

then by (i),

σes

{C\

σes

σφM

\φ

sTV} is in 21 and refines

every

. Let 58 consist of all sets

such that both V and V

are disjoint unions of set of J*f. By

(3.2) and (ii), j£fS&+

& and ^8 is an α-field. Thus, j^^J^gί?.. Since

every

set of 23 is a union of sets of jSr^, the same is true of ^~ and

particular, ^f is dense in

We now proceed to prove that ^ is not (a, 2*)-distributive. For

each pair (x, η) with xeX and ηe Y, define T

(Xt

^ — {feZ\f(x) = η}.

Clearly

iXtΎ)

e £?. For each η e Y, let A

={T

Cx>

^\xe X}. The argument

is completed by showing

(1) A

is a

2*-covering

of ^

(2 ) no covering of J^ refines

every

A,.

Proof

of (1). Evidently, Z^ = 2

, so the only thing to prove is that

the

l.u.b. of A

in J^~ is Z. The

first

step is to show that the conditions

(i) and (ii) of (5.2) are

fulfilled,

so that ^ is dense in ^. Condition

(i) is clear. For condition (ii), let L=L

w>(p

e S^. Then

L°=\J

xew

e Z\

f(x)Φφ(x)}=\J

ew(V{T(

XlV

')\ηΦφ(x)}) is a union of sets of

jSf

Since jSf is dense in .JS it is enough, in proving (1), to show that

if Le^

satisfies

LΓiT

(Xtη

)=φ for all x, then L—φ. Suppose LΦφ and

say

L=L

w>φ

Pick /eL and let #eX— TΓ. Define geZ hy g{x) = η,

9(y)

:=z

f(y) if ^^^. Then ^e

(aj>

and geL. Hence, Lf]T

,^Φφ, which

is the required conclusion.

Proof

of (2). First note that Π,

€

F(UΛ) =

For

otherwise there

would be an/e Z whose range included

every

ηeY, contray to the bounded-

ness of the functions of Z. But if A is a subset of S^ which refines

every

, then iMsuA, for all η. Hence, (J AgΞ ΓU

( U^) =

Φ,

so A

cannot

be a covering.

REFERENCES

1. G.

Birkhoff,

Lattice

theory,

revised

edition, 1948.

2. R. S. Pierce,

Disίributivity

Boolean

algebras,

Pacific J. Math. 5 (1957).

3. R. Sikorski,

Products

Boolean

algebras,

Fund. Math. 37 (1950), 25-54.

4. E. C. Smith and A. Tarski,

Higher

degrees

completeness

and

disίributivity

Boolean

algebras,

Trans. Amer. Math. Soc. 84 (1957), 230-257.

UNIVERSITY

OF WASHINGTON