everywhere-defined continuous functional, there has arisen the freed of the investigation of the maximal functionals of the spaces C a , which for ~ = (010) give ...
M A X I M A L AND E V E R Y W H E R E . D E F I N E D
FUNCTIO. NALS
Yu. L. E r s h o v
UDC 517.11
The author [1, 2, 3] has defined and investigated the class ~ = {C~ l~eT} of continuous partial functionals of the finite type over /V • In connection with the different possibilities of defining the concept of an everywhere-defined continuous functional, there has arisen the freed of the investigation of the maximal functionals of the spaces Ca , which for ~ = (010) give also the set of all everywhere-defined functions from /~ into ,¥ . In the present paper we investigate further properties of everywhere-defined functionals and the relations between maximal and everywhere-defined functionals. I, On each of the spaces Ca , 6 e T there exists a natural partial order which will be denoted simply by --~ (we note that for Co=h / the order ~ is trivial (coincides with = ) and is different from the natural ordering on /V ). The functional f e C~ is said to be maximal if it is a maximal element of the partially ordered set . The subset of all maximal functionals in Ca will be denoted by C ~ COROLLARY. For each ~ e ~ d there exists a maximal functional
Indeed, this follows from Zorn's lemma since the partially ordered set Given the functional f ~ ¢
we consider the set ~ { ~ ' ] f ~ C
such that
](E~'~}
(~,
6 7P~]=>fl~]
~)
f~,~
is inductive.
forany ]eC}.
We
note that this set is nonempty since it contains the functional ~ . Let us show that ~'~ is directed. Indeed, if f l, file ~ ~I_~ ~ and any ] e C
while ] e C ~ 7till
is such that ( _ ~ ]
(according to the corollary such a ] always exists), then
;therefore f ' and 7plr are compatible and ? ¢ ' V f 1 I ~ ] o
suchthat~ ~ ]
Sincethelastinequalityholdsfor
, we have f d {II£ ~--. Thus, ~ is directed and, consequently, there exists
s,,p.p Without difficulty one can prove the following properties of the operation. :
1. f-
Ci
2. 6,4 {, ==::>f; ~ 7£; ; A functional ?C satisfying the condition ( ~
{
is said to be closed, white the operation ~rt.,.?e*
i s called closure. The following proposition gives another characterization of f * . PROPOSITION 1. If ~ , 5 e ~ - then we have that ~ tblewith f , f, compatible with ~0 )"
/c ~ >
Proof. Necessity. It is sufficient to verify the condition for , then there exists ~,t/f and ] e C ~ suchthat ~1~]
f, o f ~
for any fo= f *
• Since
~ £ ~
( f7
compat-
If ~ is compatible with
{-~7 , w e h a v e
?e~]
,butalso
; therefore ~ and Tp* are compatible.
Sufficie.ncy. If ~ 0 ~ ' then there exists ~ e C ~ such that f ~ and f o ~ ] " Since ~ is maximal, from ~ 0 ¢ ] it follows that fo and ] are not compatible while ?e and J7 are compatible. Translated from Algebra i Logika, Vol. 13, No. 4, pp. 374-397, July-August, 1974. Original article submitted September 13, 1974. © 19 75 Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. A copy of this article is available from the publisher for $15.00.
210
The p r o p o s i t i o n is p r o v e d . Thus, ~ i s
the l a r g e s t e x t e n s i o n of 7~ which is c o m p a t i b l e with any o t h e r e x t e n s i o n of 7~ .
P R O P O S I T I O N 2. E v e r y
~ - e l e m e n t of the s p a c e
C~ is a c l o s e d functional.
In o r d e r to p r o v e this p r o p o s i t i o n we need the following: L E M M A . Let ~/i , [ ~ n , a nonempty b a s i s neighborhood Proof.
A u~
x Vj
~z
._• -
and let h / ¢
V~
V~ , ~ < n
, thenthere exists
We note that if the l e m m a holds f o r the ~ - s p a c e s X and y , then it holds a l s o f o r X x )/ .
Indeed, let V i ' ~ V / : ,
then
be a b a s i s n e i g h b o r h o o d in bo • If \~] s u c h that ~/ ~ '~ \ U V~
~ 1L,
. . . .
s-Z(a,/),'cox
We a l s o s e t
] ~n
then t h e r e is nothing e l s e t o d o . If, h o w e v e r , 6e = (0; . . . . . a]'_,lo)
~= ~t'd,~/~ ": T
(or a z = o d-
g ' = ~(a,j)< ~z such that
~
"
In order
to prove the theorem it remains to verify that ~ / is an everywhere-definedfunctional, i.e. ~u£ G~ If
~aGg
so that
, then for any
~qvf~J
E,:,
such that de=~ O,
Ae~GGc-~]e.Consequentl A D ~ u J y',~aC7 = ~
__~yv,
is defined and y y e ~ 6 .
The theorem is proved. In order to formulate important consequences of the theorem it is n e c e s s a r y to introduce a new coneept {whieh soon turas out to be old). We define for eaeh for 6= 0 the relation ..... ¢ _ , 1 o )
for
"6
f~ ~
V~, i =
Indeed, if f i ~ ~
--'6 on G~ by:
coincides with equality;
,
COROLLARY l. If /"o'~ E ~ 6 ' s u c h that
~ e S T the equivalence relation
/"o "~6 /"7' then there exist n o n o n i n t e r s e e t i n g open sets V0 and ~/#
o, 4 ,
g=O,t,
Von ~
= ~
and
~f,
,
then
6VO
and ~ = n t~ 5 ,
x U ~ } ¢" ~
n-t , ~ i.~ ; t h e n t h e r.e exist ~o ] ~-~ S} , where ~ is a basis neighborhood in 0~ : ~
D
u~- and the following conditions
hold: 1) V~ V/ , ~ , j < s ,
is either empty or it has the form
~,
~45 ;
219
~eC~ is defined by:
3) the value of the function qa at
]o
if
/,' =~i ] ¢{b~s
if
ff e U 17/ , then there exists a smallest neighborhood
, then
('~)
is the smallest element of ~o ' io < S , containing ~ and then
5'
According ~co the lemma of See. 1, we can find a nonempty basis neighborhood v
-,f
US
ha vingthe form
if
for the f - e l e m e n t
the system of pairs t< ~ , ,~i> ] _~ ) r e
~]f
and ] ;
i , s } U t} ; then
(]z} .
Let
is not compatible with
F{cCo.]r)[~).
such t h a t ~ , ~ 4 a ; . ~ , - - ~ ,' ,
f
c ;J
We consider now the ease
de'ined functional from
"
]ze
, ~,E~'
is an ~ -element and
and ] ~ (]/(~q;))=a~
]z~q, '
I
~o = ( o / o ) , ~ , , ~ z = 0
I
Since ~
. Thus, ~-.- "-'
~" , i.e.,
and ~ ~£~
: let ~. be any noneonstant, maximal, e v e r y w h e r e :
O , then, obviously, ~ E
7~ ~(om)
F r o m h e r e ]1--
consequently,
(such functionals, obviously, exist). Then
This is a partial mapping from
coincides with . ~ , ~, ~
by
Applying t w i c e t h e p r e v i o u s lemma, we find ?t _
and
tion from N into 5/ equal identically to
r", (f, ~)
Ca
and the lemma is proved.
]~-~ -~ (~to,~)(~z) ~ ]o (],(]2) ; ]j
are compatible and this contradiets the ehoiee of ] j
"-'6
~?o' ( ~ / 1 = £
~; and
Then, according to the lemma of Sec. 1, there exists an ?e -element ] ; such that
elements ~7;,]/,]I /
We define now an ~ - e l e m e n t
~,~=~f , then there exist 7e -elements % E ~
and
such that .~, (~o ~ ) ( ~ ~ < )r ( ~ , ] , )
]~< Z(]o,~,)(£) .
j~,
¢
is the func-
~(o,o(.)~-(olo~.m
We consider
~(o/o) into ~ , equal to 0 on the domain of definition which obviously we have , ~ (~,~) 4 qow,/o) " Thus the system
gives the desired example•
We proceed now to the problem of the correlations between the operations ~ , ~ and the classes Go of everywhere-defined functionals. Since ~
is closed under extensions, it is clear that
C¢
is closed
also with respect to the operations + , ~ , i.e., 3E The inverse implication for the operation ~ does not hold, as shown by the following example. Example 4. If and i.e.
f~tozo) ~ e G6
/ e ~)((o,o),o) is the functional from Example 1, then ~
' while ]~ is aneverywhere-defined mapping of :/=:>
ff((ow~/o), since / ~ / _ ~
~(om) into 0 , so that ~ ÷ a
~ffow)zo) '
]e~6
For the operation ~ of deductive closure the situation is different. PROPOSITION8. ff
]e~ 6
,then
]£~0~]'~e~ e
Proof. One needs only to prove that 5 ¢e G-6 ~ ~e~T e Cg of ]o
First we prove that such that
]o~
]'eG6---~->~?eg6.
~e ~v
. It is sufficient to consider the case
If ~e.P~?,/3~ff
and either ] is defined on £
, then there exists an
f-element
or ~ is defined on all the proper extensions
(and such extensions exist). Thus, in all cases, except possibly the case
]= ]o, ~7 is
defined on ] .
If ~7= in, then ] is an everywhere-defined f -element. From Proposition 6 and Remark 3 given after the
220
t h e o r e m it follows that ] is a maximal element, i.e., ] d o e s not have p r o p e r extensions. Consequently, also in this case ] e_P~? , and thus ] is an e v e r y w h e r e - d e f i n e d functional. The previous arguments show that _~] (?G~- = ~ ] ' n G K even for an a r b i t r a r y 9 e ~ . F r o m here, by a transfinite induction we obtain for all ordinals ~c that _D] ~ 2 ~ - - . D ~ : f} ~ff • Consequently, _py n ~ g = D 9 8 ~/Seg O ,then /_~]Sglgj=gff _P]O'~ff=Gff , i . e . , ] ~ G ~ .
n Gg,
and if
The proposition is proved. Remark.
F o r the operation
~ x
and
~EST
one can easily verify that if
III. This section is devoted to the elucidation of the relations and ~ . The class _~ and its e m b e d d m g ( S - m o r p h i s m ) M are of T h e o r e m 1 [1, See. 5] it follows easily that Fo (~o) ~ 0~ f o r ent p a p e r shows that ~'~ c~ /l (/_7~o~0)/o)) ' i.e., /2 ( ~ t o w ) m ) ) ~ ((o/0)]0) ~ definitions it follows that ~ (D((o~o)/o~)= Gc'o~o)'o) /~ Cm
((010)10)
.
~ - ~ X , then
whieh exist between the L -models .~ defined in [1], See. 5. F r o m the proof 6 ~ S T , while Example 2 of the p r e s t'R ~(to/o~o) since f r o m the corresponding
F r o m all the r e m a r k s made it follows that
(under the natural identific ations): .Do= go , ~ ~mo) = ~ (ore) ' /-ff((o,o),o)= ~(to,o),o) and ~¢om),o~m) ~ ~ (tw, o),o),o) " Let us prove now that the last inclusion is p r o p e r . Example 5. If 6 " = (((o/o)/o)/o),
that
F, ~(~ . In Co
6= ((oto)~o)
, then t h e r e exists a functional
t h e r e exist f - e l e m e n t s which are maximal functionals.
defined by: ~o(]) --~ 0
for all ] e C(o,oI , in p a r t i c u l a r ,
of
is the unique element of the neighborhood
C (0/o)
, and
Fe
F e / z ~ . (/_06,) such
For example, F o ; C(o,o) ~ A /
F o (~') = 0 , where
is
0" is the s m a l l e s t element
. We construct
F, in the fol-
lowing manner: f o r H e C 6
5 OH} ~
Obviously,
I
o, if Y = 5 ; ¢ , if /-/and 5 a r e not compatible; not defined, otherwise. /r/
~ e C6..
F, e C~# • It can be verified without difficulty that
o r d e r to prove that
F, ¢ ~ f
we construct the functional
Fo'e ~
and
F,, e/z (_~6~).
In
.
{ O, if ]¢-0"; _ not defined if ]=@. Clearly, ~1eff~
, but F is no, defined on F : .
Thus
F7 e/u (/_76 ~ ) \ C~ ~ and ~ is the desired
functional. IV. In this section we will show that the class K l e e n e - K r e i s e l funetionals [6, 7]° THEOREM 2. The class ~ of K l e e n e - K r e i s e l functionals.
$ of e v e r y w h e r e - d e f i n e d f u n c t i o n s coincides with the
can be identified in a natural m a n n e r with the class I ( = ~K~ I o E T}
Proof. Let P ~ 7 be defined by: p is the s m a t l e s t subset T s u c h t h a t OeP,~eP~(6zo)~P We r e c a l l the definition of the class ~ p = { / ( I o ~ P} , given by Kreisel [7], in a slightly changed f o r m in o r d e r to c o m p a r e it m o r e conveniently with paper.
~ . We make use of the density t h e o r e m found in the same
We define the s y s t e m of f o r m a l neighborhoods c4b={ % 16 C P} ation: 0 : c,b o - ~ { { n ] ] n e N } U { ~ } ;
[/"n
~ ~
f o r any
U°c~bo;
together with the intersection o p e r -
{ m } r J { r / } ~ ( "t{; }
if if
/72----//;
rn ~ n ,
221
}
.....
. . . . .
,__
.
-"--
n % ¢~
otherwise;
Uq~°) ~ ~ ~ ¢J By q~+$ we denote
if
:
Ut¢~o~ c~(e;to).
for any
q ~ \ l ¢ } • /~0 ~ A/ : the relation
¢o:
~zeo(f
for n • /~o , U ° e c~ 0 is the
usual membership relation E .
K(~,o;..~ {~ [ f is a partial mapping from gb; into N such that: 1) from t h e f a c t that U~n U ~ ¢~ and f(U ~) is defined it follows that ~ ( U ~ and ~ ( U ~ ) = ] c ( U ~ n 5 ~) ; 2) f o r a n y
If that
7£•
.7~(Vi.}
•
~e/~
there exists
f(vto)' Ut'm'
suchthat
flee
t ¢'v/' zzz > I ~--I ..... K} e ~ f ,6,o,
is defined and
equivalence relation ~-~
U*eqb:
f(~)=n~
by:
"%
forany
L/6E ~ :
Then, for
is such that
and
{ ( U ~) i s d e f i n e d ~
.
f e cato~
means
F o r e a c h ¢ one defines on /~s an
coincides with the relation =, while for ~o, ~l • Kc~/o)
We note that from the density theorem it follows easily that if 7PE6U 6
0
,thentherelation
~=4, .... K
~o"(,w,~ ~-~ VUeqb; [~o (U)and f.l(U)
that
U
U~) is defined
7e£ K"-(6,o, and
~e/~ 6
defined U6~
~----')~(U)=~(U)J. , then there exists f • / ~
w e d e f i n e 7e(~) to be equal to
~ e U ~ • This definition is independent of the choice of
f e d ~)
b/~
such
. where
It is easy to
verify that
Setting /(6--~
/~6./N
, we obtain the required mode]
We define now the mapping
~-
~ " q~
.
")
" ~o
~'p , a class of K l e e n e - K r e i s e l functionals.
in a natural manner: each nonempty formal neighbor-
hood Lr of type 6 defines a nonempty basis neighborhood in C~ ; the { -element which determines the
gg~ (U) . We denote the image of "~a (gb;) by "~6
neighborhood will be denoted by Remark. Unfortunately, smallest element < ~,/Tb
~
does not coincide with the set of all f -elements (for example, the
ff~ in g'6 for o?g0 and all the elements corresponding to neighborhoods of the form
and so on), which generates specific difficulties in the proof.
We note that the family of are compatible in C6
{ -elements
, then ~ u ~
e ~6
LEMMA. Let X be an 7g -space,
2~6 has the following property: if ~eo , x r e 56 and ~ o ' ~ . Let us prove now the following lemma.
X ! a subspaee of X consisting only of { -elements (X/--~ X0)
and such that if Xo,X, EX I are compatible in X , then they are compatible also in X / . T h e n f o r a complete
~0
-space Y and an a r b i t r a r y continuous mapping
tinuous extension
222
] * : X ---,-Y.
~ : X C--- Y there exists a smallest con-
Proof.
For the space
](:c o) ~ ( ~ )
for
X I the continuity of ~ means simply only monotonicity (:~o~zc,
). We extend ~ t o t h e monotone mapping ]o: Xo-~-Y by:
~=oe Xo
{~(xl)
. We note that the definition is c o r r e c t since the set
~lo[:Z'o)=~ Y ) o r it is directed (since
empty (andthen
i~ lexl
{x'] :c'eX',z'~xo}
, ~: I
--~ ~ o t is e i t h e r
is e i t h e r empty o r d i -
rected, while ~ is monotone). It is c l e a r f r o m the definition that ]o is monotone and extends ~ . But for
X o monotonicity and continuity are equivalent and e v e r y continuous mapping
the basis subspace
the basis subspace X o into a complete of the entire space X and
~/0 of
~o - s p a c e ~/ can be extended uniquely to a continuous mapping
Y . Obviously, the continuous mapping ~ ~- obtained in this m a n n e r extends
~/ and is the s m a l l e s t of all such mappings. The l e m m a is proved. Remark.
If
Under the conditions of the lemma, the mapping
f e/((61o~
mapping
from
%
into
5 (ff)
mapping ; ~
from
~6
into
"/=o(N) . In other words, ~ e
into
Remark.
X~,o )
According to the lemma,
E v e r y mapping of the f o r m
: if
~(51o)
~?~¢
~,
By induction on d e P
and h a s t h e f o r m
: p~"
n~;>
(~Z~
where
5~e~9~,then
~(~/o) onto ~'v
]e:TC(~w)
d e t e r m i n e s in a natural
which satisfies condition 1) in the definition of the class K(~,o)-
we will prove the following a s s e r t i o n s :
1. The image of
~a~
under the mapping
2. The image of
/~(~,o~ under the mapping
5(~,o ~ , i.e., f o r each
K-(o~o~
" Thus, the mapping / 2 .
This follows f r o m the r e m a r k made a f t e r the l e m m a and f r o m this p r o p e r t y
m a n n e r some partial function /~* on g~+
~=o
can be extended to a continuous
~ e £7r~,o) can be r e p r e s e n t e d in the f o r m C--
Conversely, the r e s t r i c t i o n of each functional /e f r o m
For
f
~(6~o~ is defined.
sup { ~o J~o= '~,o) (d), ~e U} of
can be written immediately as:
, then ~g induces (correctly, as can be easily verified) some continuous (monotone)
~
f r o m /~/~o)
~
/ z - - ~ ÷ is contained in /~(~w~ f~-~*
/~ e b---(~o) t h e r e exists
is contained in ~(~,o~
~a/~,o)
such that
~
and is co-initial in
£
these a s s e r t i o n s can be verified in a straightforward m a n n e r (moreover, ~(wo)
are identified by the inverse mappings We a s s u m e now that for some
V~o e P
+~
, 2"~
and
).
the mappings and
p o s s e s s the p r o p e r t i e s 1 and 2 (with ~ instead of (6to~) Let us prove p r o p e r t y 1. Let h a G(olo) inition of K ( 6 / o )
As mentioned above, ~ satisfies p r o p e r t y 1) of the def-
We prove that /z satisfies also p r o p e r t y 2) of this definition. Let We have to find a neighborhood inductive hypothesis, *
•'e
U¢ ~
such that
~e6U
G6 ; therefore]zis definedon ~
and
/z÷(U)
~ ~ ]~
be an a r b i t r a r y element.
are defined. According to the
. Then, according to the r e m a r k made above,
can be r e p r e s e n t e d in the f o r m of the limit of a directed family of elements f r o m
2/0
. Therefore
223
is defined on s o m e
]o~$6
such that ] a ~
such that '~6 U) = ~?~ , while
~
. But then there exists a neighborhood
U6c-P +
] e 6 i/ . According to the definition of /t+, it follows f r o m these proper-
ties that ,/i~ is defined on ~ , Property i is proved. W e prove now property 2. W e note that for all / 6 K(6m),
~@~'], e
form
is situated in
; consequently, (~" e G(6m) " Thus, we have proved that the image of
~(~m)
Let us p r o v e that it is co-initial in
k* e/
