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RELATIVELY COMPLEMENTED,

DISTRIBUTIVE LATTICES

Yu. L. Ershov

UDC 512.565.2

This paper is devoted to an algebraic exposition of the foundations of the theory of relatively complemented,

distributive lattices.

description of superatomic Boolean algebras for Boolean algebras concepts or results.

Some of the results presented here (e.g., a

[i, 2] and a reduction of the isomorphism problem

[4]) are known, but we give purely algebraic proofs, using no topological The theory of extensions for relatively complemented distributive lat-

tices was discussed by the author in 1971 at the llth All-Union Algebra Colloquium at Kishinev. i.

Category

~

Theory of Extensions

The most throughly studied class of distributive lattices is the class of Boolean algebras. factory.

However,

the categorical properties of Boolean algebras are not entirely satis-

For example, although a homomorphic image of a Boolean algebra is completely deter-

mined by the kernel of the homomorphism,

the kernel, an ideal of the Boolean algebra, is

itself not generally a Boolean algebra.

Therefore,

it seems advisable to consider a wider

class of distributive lattices that preserves most of the good properties of Boolean algebras and has better closure properties. Suppose =5

~=

; an element

CEAis

is a distributive lattice with zero, G , ~ E A , ~

called a complement of

a

relative to ~

if

, i.e., ~ N

C m ~ = 0 and a u C = S ,

Note that if a complement exists, it is unique. A distributive lattice O,~

~

£~

there exists a complement of

mented, distributive lattice, define on

is called relatively complemented

~

~ relative to g . If ~

if for any pair of elements is a relative comple-

the uniqueness of the complement, noted above, enables us to

a binary operation

~

such that the following identities hold:

t. In any relatively complemented,

Cm

distributive lattice we have:

g}\x =

z)= =(z-z); 6) ( ~ ) U (X\])=X. Translated from Algebra i Logika, Vol. 18, No. 6, pp. 680-722, November-December, Original article submitted June 6, 1979.

0006-5232/79/1806-0431507.50

© 1980 Plenum Publishing Corporation

1979.

431

Relatively complemented, distributive lattices will be considered with signature



Note that a homomorphism ~ : ~

lattices with signature in the signature < U ~ , \ ~ C=~

; then

O=

0~.

of relatively complemented, distributive

preserves the operation ~

Indeed, suppose ~ , o e

~IO) = ~ ( ~ C ) - ~

~(~)~(0),

, i.e., it is a homomorphism

]~] ~

~ = O,

~0

~ ~ u ~,

~(~)=~(~U~)=~[~UC)=

~]0

i.e.,

F ~ ) U ~(~)

and @(a)=~(6)- {(a). The class of all relatively complemented, distributive lattices and the class of all homomorphisms between them form the category lattices.

We write ~ 6 ~

to denote that

~

of relatively complemented, distributive

~

is a relatively complemented, distributive

lattice. If L% 6 4

and £~

contains a largest element, then

£7~ is called a Boolean algebra.

Boolean algebras are often considered with signature < U , ~ C ~ O , ¢ > , same meaning as in ~o

; the value of the constant

a unary operation related to -An ideal of a lattice

With any ideal J lattice of ~

modulo

U~ n

, and -,

)I,n,0

have the

~ is the largest element, and

C

is

as follows: C~ = /~

CLE~o

is any nonempty subset

of a lattice

~

7c_IOgl such that

we can associate a new lattice

OC/~ , the quotient

J .

We define on the set I~l (~ZO~=gUC)]

where

an equivalence relation as follows: f ~ i 4 a , ~ > l a f 6 1 C ~ l ~ C e J

; on the quotient set t ~ I / ~

we can define in a natural way the operators

so that

We obtain a lattice

~'/~7

, which also belongs to ~e

morphism (the projection) of ~

L21 ,' [~2_]~,

; the mapping

is a homo-

onto the quotient lattice C~/~

We now state a number of results known as the homomorphism theorems. Suppose ~ , $ ~ ~0

and ~ : ~

is the set ~ { L Z ] a E I ~ I ~ Proposition i. ~.~

--~ ~ / ~

--~ $

~/Q)=0}

is a homomorphism.

The kernel of the homomorphism

It is easy to see that

Any homomorphism ~:C~ -~ ~

If ~

is an ideal of ~

respondence between the sublattices of

~/~

~

is an ideal of £~

can factored into a product of a projection

~a)----[~]~z ~ ) and an isomorphic embedding ~ :

Proposition 2.

~

~/~.~

) ,,~

{~=~, ~).

, then there exists a natural one-to-one corand the sublattices of ~

containing

~

; more-

over, ideals correspond to ideals. Proof of Proposition i. there exists 0 ~ ~(ao) = t~(ao)u o

432

We first define a homomorphism such that

~=~C=~,t~6 ;

= p~a,)) u p ( c ) = ~ ( a , , u , " ) -

~,:~/~

" ~.

If

).

= ~,

this mapping is an isomorphic embed-

It remains to show that the mapping :is onto.

CoEgg .Cl~a"L, then for d~CoUC we have O&n~= O~n(CoUCI)- (~ZnCo)U~2R07)~ CouO=C o ~ u & g = ~ u ~ . Therefore, the image ~ ( z = C I , since C1r]{;=O,C1[J~l=c1uIcouo.)=~Cot]c1)~i~Z= is



and of

.

We will say that g~

is an extension of

g~/--~Po is isomorphic to ~;

~o

by

,-..~°

~',. ,~

in

if ~o

is an ideal of

~

and

~,- /)~ ---,-0.

E x a c t n e s s o f a s e q u e n c e o f homomorphisms ~ 2 - - ~ ~

~,

Extensions will be identified with exact sequences of the form 0

morphism

If

is the kernal of homomorphism

~

~ A• ~ (In?=&

(*) means that the image of homo-

~).

An extension

is called equivalent to the extension

(*) if there exists a homomorphism

£ :~

~=~)~ such

that the diagram

o--is commutative.

.

E~

( ~ )

~

that

diagram

434

~

is an ideal of ~ !

following condition is satisfied: an ideal of ~

by ~!

is an isomorphism between ~

, to within the above equivalence,

and =~0. is denoted

We will give below a detailed description of this family, from which it

will follow, in particular,

Suppose

.z),--.-o

It is easy to show that in this case $

The family of all extensions of ~ 0 by

,dp I

(gOo,~I)



if ~

can be regarded as a set.

We will call =0 1 an ideal completion of ~ and ~ } ~ - ~

if the

is an arbitrary embedding of ~

, then there exists one and only one homomorphism

~'~--~

such that the

as

is commutative (the lower arrow is an embedding of ~

into ~fPt ).

!

Proposition 4.

Suppose

/

is an ideal completion of '~'~o

~0

' and ~ : ~ ~ / , o ~ o

there is a one-to-one correspondence between the homomorphisms ~ : ~ - - ~ 7

Then

and the extensions

(*) (to within equivalence). Given an extension (*), there corresponds a unique homomorphism identical on ~

Factoring ~

and e~); by

~o

establishes an equivalence of extension and into

= that is identical on ~)0

~I':~--~;,~':~I--~ homomorphism

6

7

that is

, we obtain a homomorphism ~'" .~ -----.~); .

Thus, we have associated to the diagram (*) a homomorphism ~t

~/: ~--.-~)o

~E

Hof~I~1,~o )

If

g:~

.

lJ. ~)J-~-o~; is the unique homomorphism of

, then ~u& = ~

and the induced homomorphisms

agree, since under factorization of ~

and

is sent into the identity mapping of ~)I onto itself.

g~l by ~

the

Thus• our correspon-

dence defines a mapping .

We will show that this mapping is onto.

Suppose ~ : ~ t - - ~ 7

sider the sublattice

~,

where ~ : ~ - - ~
E ~ @ ~ ,

"

is onto.

~IE$~)0

I .

E

Indeed, suppose ~ : ~ ,

Clearly, the image of ~

and E Z

~d~ }

isomorphic to g~o [~'~

The kernel of the homomorphism ~

then p d ~-- 0 , hence

l¢'d!

Con-

It is easy to see that ~

onto _T ) and that ~ ~ a ~

is the projection on the first coordinate. ~

by ~0

,do

contains the ideal Z ~ { C O , ~ > I ~ E ~ o }

is an isomorphism of ~Do

since the mapping

is a homomorphism.

.

~

is all of ~!

is obviously

Z,

,

since

Thus, we have an exact

sequence

It

is

easy

to

see

that

We will now show that if ~* extensions.

To an element

the

homomorphism

into

correspondS_rig

to

(**)

aesOP

we associate the pair < ~ a , ~oa>

Denote the correspondence

We can show that < ~ , ~ > o, ~ < ~ , ~ >

morphism establishing the equivalence of extensions

by $

(*) and (**).

if to two extensions corresponds the same homomorphism

~*:~?---~

, where E ~)~,

~:~--~=~;

can be identified with

This is clearly a homoIt follows at once that , then these two extenThus,

~OITLC,,~=~o),

To complete the description of E~[g~0,-~I) ~o~0

is

; but ~ z = ~ 9 # ~

sions, being equivalent to the same extension (**), are equivalent to each other.

~ (~o,~i)

is

corresponds to the sequence (*), then (*) and (**) are equivalent

the homomorphism that is identical on ~ o by definition.

of

it now suffices to show that for each lattice

there exists an ideal completion (the uniqueness of which follows from the defini-

tion). For any distributive lattice ~ ~)

the family of all ideals

~

(with zero, but not necessarily in ~0

) we denote by

(including the "improper" ideal• the lattice ~

itself).

This family is again a distributive lattice under the operations ioU~--~{aU~ laE/o , ~ E77

435

e

and /'0N/1 ~ T O N ] 1 into

(set-theoretic intersection).

m(~.n~

certain

~

is closed under the operation

relation and the fact that any element of for

and ~]

there exist elements

We will show that t(',~,

~,~'~f~,

, which is equal

lies in the lattice generated by the elements ~O~o,.., a o ~ ,

denote the distributive sublattice generated by the set of elements

~6~f

as

; it was shown above that

~of =g.~kJrt~°f£

LEMMA 2.

I)

~,

then it is easy to see that to ( ~ )

, then

is finite, it follows that

~ ~ ~0

for an ordinal

0% we have ~ ( ~ o ) ~ o

image of ~ o

~,~--f (~o ~-~o/~'~)Of]qPd~(~)

in

Suppose ~o--=~~ :

=o~)a/(P~(~o) and

inclusion ~ ( ~ o ) ~ f ] ~ 9 ~ ( / ~ { ) ment S e ~ O

and use transfinite induction.

(~(Cl-~C~)=~(~)l-I~(gg)--~fT~a(O~)={) and O- , E C

~' \~' for

It follows from condition (i) for a weak semicorrespondence

,t7

, then

I

-,~ .

~-

Putting

'

--

(L'

:

') ~

I 9~,,

,7~z

,1,

we see that conditions I) and 2) are satisfied; condition 3) follows from the obvious relation

Cn ÷ , = m ( ¢~. , n(~z~, ,-- a.~ ) ) u ( e,~ . , . . ~z~ ) . Putting

L ~9~;

~--~?,em , we see that

is a monotone mapping of ~;~o---~

~

into

such that ~#/.----~

evident from the construction of

~

That

/-

is a linear basis of the lattice m O

and

; by Proposition i, there exists a homomorphism F

satisfies the conclusion of the proposition is

L~

and ~ , , ~ W .

If a weak semicorrespondence C

between ~ o

The proposition is proved.

,~,0 > 6C~=~

= 0 , then

COROLLARY. homomorphism

~

C

and

~

also satisfies condition (i*):

is called a semicorrespondence.

If, in the hypothesis of Proposition 3,

C

is a semicorrespondence, then the

is an isomorphic embedding. 445

A semicorrespondence and

~4

C

between

~o

and ~

is called a correspondence between -~o

if the following condition is satisfied:

(3) if ~(Zj~>E£, 6 ~ f

Proposition 4. countable, then

C~_.,~o"

If

~o

, then there exist

and ~

~)~

CO,£¢6~ O

is a correspondence,

such that ~ ( ~ F ] C 0 , ~ n ~ > , < ~ - C o ,

and ~ o

~)4

and

are at most

are isomorphic.

The proof is analogous to that of Proposition 3. We give some applications of Propositions 3 and 4. I.

If ~ I

is an atomless nonzero lattice, then any at most countable lattice ~90 @ %

is isomorphically embedded in ~

C~[J[J{I

Indeed, let element in

~}

and ~ /

If

~o

' ~/

C

d ~/

and

~

is not the largest

is a semicorrespondence between ~ o

is isomorphically embeddable in

and ~

~4

are countable atomless lattices having no largest element, then ~ 0

are isomorphic.

×0"~),/

Cc-~-~)0

The set 3.

0 ~

It is easy to verify that

By the Corollary of Proposition 3, ~ O 2.

0~.~o~

If ~ O ~ ) ~

Suppose

defined above is a correspondence between

are countable atomless Boolean algebras, then ~ o

£g is the largest element of

~i

~)0

and ~ f

and ~ f

are isomorphic.

; if we put

C

then

C

is a

{, eo, e,>] U : laEmo, #O, correspondence between '~)o and ~9,

We now turn to superatomic lattices. Proposition 5. l) ~

For a lattice

~

~

the followin Z assertions are equivalent:

is superatomic;

2) each epimorphic image of ~ 3) each sublattice of ~

is atomic;

is atomic.

The Corollaries to Lemmas 2 and 3 show that assertions 2) and 3) follow from i). Consider the epimorphism If 2) holds, then

~9/~(~9)

~

--~L~)/~(Ag~;

must be atomic.

~=~O~{'~)) ; therefore, in this case ~

is isomorphically embeddable in it.

~

in

can be extended to a homomorphism 446

cannot contain atoms.

If ~

i.e.,

i). is not superatomic,

then

As was noted above, any at most countable lattice

Suppose

~:Y~o---~/~(~)

of a free lattice with a countable set of free generators are such that the image of

Thus, 2) ~

by contradiction.

is a nonzero atomless lattice.

.,~)./C~{/~))

This is possible only if ~ 9 / ~ ( ~ ) = f O ] ,

is superatomic.

We will prove the implication 3 ) ~ 1 ) ~/~(~9)

the lattice

~/~(~ ~:y~o---+~

is an isomorphic embedding

{O.v~g~,...~. Suppose

is ~(~.m]~ fg ( ~

.

4,4,"" E m

The mapping ~.~--*~, f Z ~ ,

of the free lattice; since the mapping

~

is

obviously the composition of

W

be an isomorphic embedding.

Consequently,

As was noted above, ~ o

and the p r o j e c t i o n ~ ~{j~o )

--~)/~(~,

it follows that

is a sublattice of ~

is an atomless lattice, hence

~

F

must

isomorphic to J ~

does not satisfy condition 3).

We will define the concept of atomic type for elements of a superatomic lattice ~) If O#O~,Z), Note that ~ ^ 6o

then ~

is a superatomic lattice; let ~ = J ~ { ~ )

cannot be a limit ordinal.

for all ~

, i .e. , d h / ~ ( ~ )

Indeed, if ~

be the atomic rank of

were a limit ordinal, then £,~/~)~

, and then ~ ( ~ ) = ~ ( ~ )

and ~ D ( ~ ) ~

Let

^

be the predecessor of ~ , i.e., ~ = ~ * f ~D(~) = ~

, the image

~,* of ~

in

of a finite number of atoms; let ~ since g g ~ # O [ ~ # ~ ( ~ .

, so that ~/~*(~)

~(~)#~

lies in ~ ( ~ / ~ ( ~ ) ,

The triple ~ ( ~ } = ~ ; 0 2 / z >

o/,=~o

+¢=~o,

and different from zero.

then ~, g ~

Let ~

atomic type of the lattice ~ ¢(~) :

~C~)

LEMMA 4. ~,4~,~,~

for

is the join

of &

;

as follows: suppose ~ o - ~ ( ~ ) . has a largest element.

, then by the atomic type of ~

If ~I .

gg

, then o~ = ~

,

is the desired ele-

~

~ and fg~= (7 , or satisfies the con-

(and then ~/=~

); the lattice ~ P

be one of the preimages of the unity of g~#under

Then ~ ) =

,

containing exactly

R1

~ ...., ~ ; suppose CE~

is one of

since .~%=~O) ~+~C~) ~

atoms.

Therefore,

and

g-(g±)-- ~C.

Therefore, ~.

~ ~±)

, we see by Lemma 4 that ~'(~C-~)~)=

~((~uc)±)=~f~d~)~ ) ,

Analogously,

Condition (2) is satisfied.

Condition (3) is symmetric to (2), so we omit the verification. To prove the proposition it now suffices to invoke Proposition 4. To consider the isomorphism problem for arbitrary countable lattices in 4

we must

study several technical concepts. Suppose ~ E

~

; by the atomic type of the lattice ~

and by the atomic type of an element denote by ~ I ~

Proposition 8. ~+/

we mean the triple ~(~)~-~ ~'I~n ~

~(=.~) (~(d)); we

(~{~) the middle term of the triple

special rank of the lattice ~

into

~ ~~

If

(the element ~

g~)~0 , ~ = ~ ( ~ ) '

we mean the triple

call

T~)~(~(~)), (~)).

We

~(~)(~(~))

the

).

then 6:~i

• @(~), ~ £ ~ ,

is a mapping of

such that

6(o):o, It is clear that 6(0)=0 and m f d ) ~ D ( o ~ ) = ~ then the ideal ( ~ o ~ # ) f ] ~ ]~_~{] ~

~)

for any

~e~

Suppose @~o)~D(~#) ;

can be represented in the form

(~(~))÷],

where

(~) , hence, by Lemma 4,

Consequently, ~ ( ~ ) > ~ f N ~ { ~ d 0 ,

~

; if J~---~(do)(~(d~))

, then in D#--- ~)/@j[~)

the

images of the ideals ~i~90~(~)~ b=G,/ , are principal, hence the image of the ideal ~ ^ (~o~ ~ ) { ] ~ ( g ~ ) ) = ( ~ o ~ (~))~ ( ~ ~(~)) is also principal; therefore, @ ( ~ ) ~ = ~

t~4,~#}

; it follows that ~ ( ~ 0 ~ ) = ~ 2 5 g

A function Z:~--~=f+/ Proposition 8 (for

is an ordinal, that satisfies the condition of

@ ) is called additive.

A superatomic lattice LEMMA 5.

, where ~

t~o,~l~

If ~

~

is called special if its type is •

is a special lattice and ~6.~

, then

¢(~)-

@(~)

This follows easily from Lemma 4. COROLLARY. element

d~

LEMMA 6.

If ~

is a special lattice, then

and ~ ( ~ ) = ~(d±) • If

~o,~;

are special lattices and

then it follows from @ ( ~ o ) = ~ ( ~ : ) ~ i ) - - -

Remark. and

d ± is also a special lattice for any

If ~,

and ~

~I, ~ /

~('~/~ and

are special lattices, then

@(.,f~o+,,~#)--m~2~{@("~o),~(~,)}

are arbitrary lattices in ~

~'(~o+~'= ~(~-~/)

~,+~'Pl

that ~(~i)

is also a special lattice



449

,

Proposition 9.

If

~

is a superatomic lattice, then ~@

able in the form =~==o~ o + ~ and

~7

, where

is either special or represent-

~ 0 is a Boolean algebra of atomic type < ~ 0 , fg >

is a special lattice such that

~

oC~)(=@I~I)

.

Indeed, suppose the atomic type of

~

is

If ~=0

lattice Co~-~)

If N ~ O

, consider the Boolean algebra

an element ~

that is a preimage of the unity of

~P

~:~-~).

Then, as is easily seen, the decomposition

, then ,~

~P--~/qb

CA)

,

is a special

and choose in o~

(under the natural projection ~=Tq-~

~

is the desired one.

We will now prove a technical lemma that is important for what follows. LEMMA 7.

Suppose ~

~0~,~,~ t

(so that ~ =

~(o.~ --- ;

is a lattice in ~o,

).

,

Suppose

a r e

ae~,6(a)=~ ,6(~t)=~

let

triples such that:

1) 2) if

~;



Proposition i0. i) If ~ )

(

and

If r ~ = O

C#

in

~

Assumsuch

, put ~ - ~ 0 , ~ / ~ - ~

Co#d ,C?~

Now put

also, it is easy to see that

, the images of (~

Suppose

=

~ )

and

~

in


,/ of

such that G-~< .LJ~

Let C~.....C £ ~

b=q

C~=~4a+d

~o,,..:

are atoms of ~

and ~a,f~ ~

and a ~

~

its image ~

for any agog in the lattice

g~t2

atoms of

~0¢

different from the atoms

be such that ~ =

~./-~ 0- , the atomic type of i(/zf) ,

there exist elements

~£~.

such that ~ f ( ~ )

Put

1_ o then

(~)

~.~1~...

,¢{¢,,.,:,,',+,...};

is the image of f'

, then it is easy to see that / ( ~ ) = ~

(since

the sequence ~o-~2. / , we see that ~(~0-dl= ~ . depend on ~o~/~

d~),~}= ~I (4) ;

~

~=~

, there is nothing to check.

Let If

~ta2~{Z(~o~),/]= ~(~)= ~