A C T A U N I V E R S I T A T I S L O D Z I E N S I S FOLIA MATHEM ATIC A 7, 1995
Marek Balcerzak and. Stanisław W roński
O N c r-ID E A L S W I T H O U T M A X I M A L E X T E N S I O N S
We ch ara cterize th ose <r-ideals in a Boolean tr-algebra w hich have no m axim al ex tension s in th is algebra. We show som e ap plication s.
It is well known th a t every ideal of a Boolean algebra is included in som e m axim al ideal of th a t algebra. It seems in teresting to verify if th e above fact has its analogue for er-ideals.
Let us observe first th a t, if A is a m axim al a-ideal in th e class of all cr-ideals of a given cr-algebra, then it is also m axim al in th e class of all ideals of th a t a-algebra.
Really, if A is not m axim al in th e class of all ideals, th e n the re exists a m axim al ideal A ' such th a t A $ A '. Let a be an elem ent of o ur cr-algebra such th a t a 6 A ' \ A . T hen the cr-ideal generated by A a nd a, being a p rop er extension of th e a-ideal A , cannot be a prop er ideal. T his yields th a t there exist elem ents a i , a2, . . . of the cr-ideal A such th a t 1: = a V sup a¿. Now we conclude th a t —a < sup £ A a nd —a € A C A '. T h us we ob tain th a t a £ A ' a nd —a € A '. It is im possible.
Now we introdu ce the notion of an essential ideal. We shall say t h a t a p ro pe r cr-ideal A of a given <7-algebra is essential if a nd only if for each m axim al ideal A ' including A there exists a sequence ( a ¿ ) ^ x of elem ents of th a t cr-algebra fulfilling th e conditions:
(1) a.i £ A ' for i = 1 , 2 , 3 . . . , (2) inf ai € A.
T h e o r e m 1. A proper cr-ideal o f som e cr-algebra is essential i f and only i f it is no t included in any m axim al cr-ideal o f this cr-algebra. Proof. =>• Suppose th a t an essential cr-ideal A of a cr-algebra A is included in som e m axim al <r-ideal A* which is also a m axim al ideal of A as we have noticed above. According to the definition of an essential ideal th ere exists a sequence ( a , ) ^ j fulfilling b oth conditions (1) a nd (2). Since a, £ A ', therefore the cr-ideal A'a . generated by A ' an d a{ is essentially larger then A '. T h us 1: € A'a. for every i because A ' is a m axim al cr-ideal. It enables us to conclude th a t for every i th ere exists &,• G A ' such th a t 1: = ai V bi. As a result of tak ing (2) in to account we ob ta in 1: = inf (a* V &, ) < (inf ai) V (sup bi) E A '. It is im possible since A ' is proper as a m axim al ideal.
<v= Assum e th a t a prop er cr-ideal A is not included in any m axim al cr-ideal. Let A ' denote a m axim al ideal containing A. Since th e cr- ideal generated by A ' is not proper, th ere exists a sequence ( a * ) ^ of elem ents of A ' such th a t sup «, = : 1. This enables us to conclude th a t —ai A ' for each i, a nd th a t in f(—a ,) = : 1: — (sup«¿) = : 1 : — : 1 = 0: € A. It m eans th a t A is an essential cr-ideal.
T h e o r e m 2. F or each a-algebra A o f subsets o f [0,1] containing all Borel sets, a a-ideal 2 o f A is a m axim al a-ideal in A i f and only i f it is o f the form
(x ) = { E C A : x i E } for some x E [0,1].
Proof. =$■ C onsider two cases:
1° J contains all singletons {x}, x € [0,1]. The n I is an essential cr- ideal. Indeed, let A be an a rb itra ry m axim al ideal A of A . We define a descending sequence of intervals as follows. Since A is a m axim al ideal, therefore eith er [0, \ ) ^ A or [ § ,l] ^ A. P u t A \ — [0, | ] in th e first case and A \ = [ | , l ] in th e oth er case. Suppose th a t
we have defined A n = ^±1] ^ A where k 6 { 0 ,1 , ... ,2 " - 1}. C onsider th e pa ir of intervals [ £ , |£ £ f ) and A t least one of these intervals does not belong to A . We choose A n+\ as th a t interval. T he n th e set P i n ' L l * s a singleton, hence it belongs to X. T his shows th a t X is an essential cr-ideal. So it cannot be m axim al, by T heorem 1. T hus the case 1° is im possible.
2° T h ere exists {x} ^ I . If there exists y / x such th a t {y} ^ X th en I is not m axim al since the a-ideal X{y} generated by X an d {j/} is prop er and larger th a n X. So {x } is a unique singleton w hich is not in X an d th u s X = (x) since (x) is the biggest prope r er-ideal which does not contain x.
<= Obvious.
C o r o l l a r y . Each o f the follow ing a-icleals:
• the a-ideal Co o f Lebesgue null sets in the a-algebra o f C o f m easurable subsets o f [0,1].
• the a-ideal Bo o f the first category sets in the cr-algebra B o f subsets o f [0,1] w ith the Baire property,
is no t m ax im a l (in fact, it is an essential a-ideal).
Rem ark. Note th a t Co an d Bo can be m axim al tr-ideals in som e no n-triv ial subfam ilies of the fam ily of all ideals of C a nd B, respectively. Namely, consider the fam ily T of all <7-ideals A in £ such th a t
(*) (V A e A )(V Z ?c A ) ( B € C)
T he n Cq is th e greatest a-ideal in T . Indeed, let A € T and suppose th a t A € A \ Co. It is known th a t A contains a nonm easurable set B (see [1]). Hence (*) is false, which contradicts A € T . C onsequently A C C0. T h e category case is analogous.
Re f e r e n c e s
M. B A L C ER ZA K AN D S. W R O Ń S K I
Marek Dalcerzak i Stanisław Wroński
O < 7-ID E A Ł A C H b e z m a k s y m a l n y c h
R O Z S Z E R Z E Ń
Scharakteryzow ano <ridealy w dowolnej cralgebrze Boole’a, k tó -rych nie d a się rozszerzyć do <r-idealu m aksym alnego w tej algebrze. P o dan o kilka zastosowań.
In s titu te of M a th em atics Łódź T echnical U niversity al. P olitechniki 11, 1-2 90-924 Łódź, P o la n d
