class of distributive lattices that preserves most of the good properties of ... Relatively complemented, distributive lattices will be considered with signature. .
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~),/]= ~(~)= ~