Л С Т Л U N I V Е К S I T A T ! S L О Г) Z I E N S I S FOLIA M AT IIE M AT ICA !), 1907
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O N P R O P E R S U B U N I V E R S E S O F A B O O L E A N A L G E B R A
Let. D be a b o o le a n a lge b ra w it h t he u niv er se H a nd let. F \ , F-> be dis tinct, ill trail Iters o f D . T h e n t he set o f t h e fo rm { j 1 6 В :
F\ П F'j(~\ { л \ ->.)■} ф 0 } is a m a x i m a l p ro p er s u b n n iv e r s e o f В w hic h w e
s h a ll call a ba si c . s ubuni ver s e. W e pr o ve t ha t e v er y pro per s u b nn iv e rs e o f В is an in t e r se c t io n o f a f a m ily o f basi c s u b u n ive r s e s . T h is im p lie s t h a t b as ic s u b u n iv e r s e s are p rec is ely m a x i m a l p ro pe r s u b u n iv e r s e s o f a b o o le a n a lg e b ra . T h e s a m e fact, pr ov e d in a n o t h e r w a y can be f ound in [3].
For general algebraic’ background we refer the reader to [1] and for boolean algebras to [2]. In order to sim plify notations we use th e sam e sym bols for boolean algebras and for their corresponding universes. If
В is a boolean algebra, Л* Ç В and b £ B . then by ->дЛ and (Ь]в we
denote the sets
{ ->.r : £ .V} and
{.r £ В : .r < h},
is a subuniverse of B . T he subnniverse of the form F U ~<uP will be further denoted by B \ F . Recall that a filter /? of F is an ultrafilter - i.e. a m axim al proper filter - if and only if, for every /> G В , we have
T his fact easily im plies tha t, for every proper filter F, the equality
В = B \ F holds precisely in the case when the filter F is an ultrafilter.
W e will deno te the setth eoretical operations of difference and sy m -m etric difference by \ and Д , respectively.
W e start with th e follow ing auxiliary proposition.
T w o U lt r a f ilt e r s L e m m a . I f В is a bo olean algebra, F \ , F2 are
u ltrafilters o f В a n d a , b G В , then
F\ П F-i П {»i. -m . I), ~>b, a -r />, ~>(a -j- b)} ф 0.
Here -r deno tes the operation o f sym m etric difference, i.e.
a - b = (a — h) V (l> — u),
where — and V denote difference and sum , respectively.
I ’rooJ. Suppose that
Г] П F-i П {« , -ia. b, —»/>, и 4- b. -i(a -j- />)} = 0. If a H- b G Fu then
{a — /), /) — а ) П l'\ Ф 0.
If « — 6 G F ], then
- ’/> G Fi and, therefore,
-i« , />, -i(rt -i- b) G /*2, which is im possible because
W e have shown that
« -7- b ^ F\ and this m eans that
~'{u -г- b) £ F\.
In the sam e m anner we can infer that ->(,f -f b) £ F2 which g iv es us that
->(a ~ b) £ F\ П F2,
thus, we get a contradiction.
By a basic subuniverse ol a boolean algebra H we m ean any sub-universe o f the form
where F \ , F, are distinct nit rafilters o f B . Recall that
В | ( /', П F2) = ( F\ П F i) U - й ( / ? 1 П F i) = {.r 6 В : Fi П Fi П {./•,- .* } ф 0} = В \ ( F XA F 2).
Let us note th e iollow ing proposition.
L e m m a 1. E v e ry basic su b n n iv e rse o f a boolean algebra is a max-im a l p r o p e r su bm iiv erse.
Prooj. Suppose that И is a boolean algebra and F\, F2 are distinct
ultrafilters i>l В . I hen l \ П F> is not an ultrafilter and consequently the subuniverse of t he form
В Д п / Ъ )
is proper. Now we pick an elem ent b £ B \ ( Æ |(F, П F2)) and we will show that the algebra genarated by the set
(/•, n F 2) u { b }
generates B. Indeed, by I wo I dtra.fiIters L em m a it follows that, for every а £ В \ ( B \ ( F , П F2)),
T h is m eans tha t every such a can be expressed in term s of generators because
ft ( ft -f- «■ ) — “ »ft -r (-»ft 4- ( i ) — (i .
L e m m a 2 . I f A is a p ro p e r su b n n iv e rse o f a boolean algebra B , th en for e v e ry ft € B \ A th ere e xists an ultra filter F o f A such th a t
(1>]в П F = П F = 0,
Proof. Suppose that ft G B \ A is such that, for every ultrafilter F
of A , we have
(ft]« П F Ф 0 or
( “'ft]« П F Ф (Л,
For every i|| trafi Iter F of A , we pick an element, (ц? G F such that
ар < /) or <//■• < —ift and we define a subset Q of A as th e set of all
<li?obtained in the above m anner. We aim at show ing that Q can be
ex tend ed to a proper ideal of A and thus we have to prove that every finite subset of Q has a non-unit suprem um . Suppose th a t, for som e finite .V Ç Q . sup(.Y) = l . Put
A', = (ft]« n A and A'a = H > ]« П .Y. Since A' Ç (ft]« U ( “»ft]«, then A = A , U A j and thus
i = su p (A ) = sup(A 'i) V sup(A^), where sup(A 'i) < ft and sup(A^) < -»ft. Now, we get that
b - snp( A', ) =
( /> - s u p (A i) ) A (siip(A'j) V sup(A'2)) =
(( b - sup(A'i )) A sup( A'i )) V ((/» - sup(A 'i)) A sup(A^ )) < (b — sup (A 'i)) A - ib= o.
T his would m ean tha t snp(A 'i) — b which is not possible since A \ Ç A and b E В — A .
W e hav e shown that every finite subset o f Q has a non-unit supre-mum which im plies tha t the set ->AQ can be exten ded to an ultrafilter of A . This, however is a clear contradiction because - by the definition of Q - every ultrafilter o f A m ust contain the com plem ent o f an elem ent o f ->a
Q-T h e o r e m 1. E v e r y p ro p e r su b u n i verse o f a boolean algebra is an
intersection ol ;t fa m ily o f basic sub u niverses.
Proof. Suppose 1 hi»f .1 is a proper subuniverse o f a boolean algebra.
В and b E В \ A. YYe need only to show that ft does not belong to som e
basic subuniverse o f В containing A . By Lemma 2. we g et that, there ex ists an ultrafilter F of .1 such that
( b ] H П F = (->/)]д fi F = 0. Let F\ , F2be ultrafilters of В such that
b £ F \ . -t b f' 2 and F С F x D F2. Then we have b i B \ ( F \ П F-i) and B \ { F } П F-i) D A \ F = A , as required.
T h e o r e m 2. M a x i m a l e le m e n t s o f th e set o f all p ro p er su b u n ive rse s
o f a boo lean algebra are p recisely all its basic su b u n iv erse s.
Proof. B y Lem m a 1, it follows th at all basic sub uni verses are m
axi-mal proper subuniverses. To prove the converse inclusion observe that every proper subn niverse m ust be contained in a basic subuniverse - by Theorem I. T hus a m axim al proper subuniverse m ust be equal to the basic subuniverse co nta ining it.
Re f e r e n c e s
[1]. (J.CJnitzer, U n iv e rs al A lg e b r a , 2-i kI e d it io n , Sp r in g er - V e r lag , N ew Y ork, 197 9. [2]. R .Sik ors k i, B o o l e an A lg e b r a s , S pr ing e r- V e rlag , Be rlin, 1964.
[3]. J .D o n a ld M o nk w it h t h e c o o p e r a t io n o f R o b e rt B o n n e t , H an d b oo k o f B oo l ea n
A l ge br as , vo l. 2, N o r i h - H o ll a n d , 1989.
S la v isla in I Vronski
O P O D A L G E B R A C H A L G E B R Y B O O L E ’A
Niech В będzie algebrą B o ole’a z uniwersum В i niech
F u Fi bed a różnym i ultrafiltram i B . W ówczas zbiór postaci
{.)■ £ В : Fi fi Fi П {.г, -i./'} ф 0} jest m aksym alną podalgebrą В która nazywać będziem y podalgebrą bazową. Udowodnim y, że każda właściw a
podalgebrą В jest, iloczynem rodziny podalgebr bazowych. P ozw ala to stw ierdzić, że podalgehry bazowe są w szystkim i podalgebram i m aksy-m alnyaksy-m i algebry B o o le’a. Ten saaksy-m fakt jakkolwiek dow iedziony w inny sposób m ożna znaleźć w [:{].
I n s t it u t e o f M a t h e m a t ic s Łó d ź T e c h n ic a l U n iv e r s it y A l . l ’oliti'c hnik i 1 1 ,9 0 - 0 2 4 Łód ź, P o la n d
