universal recursively enumerable boolean algebras

UNIVERSAL RECURSIVELY ENUMERABLE BOOLEAN ALGEBRAS

S. S. Goncharov UDC 517.15

The question of existence of universal recursively enumerable Boolean algebras is studied for the classes of reeursive atomic, recursively enumerable atomless, and recursively enumerable atomic Boolean algebras. A negative answer in all three cases is obtained.

In definitions from the theory of constructive models we will follow the book [I], in theory of Boolean algebras [2], and in the theory of recursive functions we follow [3].

We denote by N the set of natural numbers, by ~(N) and ~f(N) , respectively, the sets of all subsets and all finite subsets of N.

We define a partial order ~ on N and functions P, S, R, L, H, p as follows [4]:

P(n) -~ [n + t /2] , R(n) -~ 2n + 2, L(n) ~ 2n + t,

{ S(2n+2)~r2n+l, { H(0)~0 , S(2n+ 1) = 2n+2, tt(n+ 1) ~- H(P(n+ 1)) + I,

p(n, O)-~n, p(n, g + t ) ~ p ( p ( n , g)),

x ~ y ~ > ~ l y - - p(x, Ol= 0. i=0

Define E,~{n]H(n)=i}. D~-N is called a tree if for each x~D and g~N from x~-<g follows y~D and S(x)~D.

Assign to each i~N an infinite subset A i of N in such a way that Ao=N, A~=ARd) UAL(~) and ARc~flAL(i~=~ for all i~N.

Taking the Boolean algebra <~(N), U, A, C, ~, N> , we consider its subalgebra ~, generated by the elements A, i~D, where D is a tree [4]. It is readily seen that X~I~o[ ~=~ (X----~ V(Nil...ik~D) X=A~IU ... WAsh). If D is a recursively enumerable tree, we define a

numeration ~ of the Boolean algebra ~,.

Let ?:N-+~(N) be the GSdel numeration of all finite subsets of N and y(0) = r Let

fD:N § D be a recursive function. We define a numeration vDPutting ~D(n)--~ U AfD(O It is onto i~?(n)

easily verified that v D is a constructivization of the Boolean algebra ~Jo and does not depend, up to an autoequivalence [5], on the choice of the function fD.

Note that there exist partial reeursive functions eD, h D such that hD(n) is defined for n~D and VDhD(n) = An, and eD(n) is defined if ~D(n) = A k for some k and eD(n) = k.

Let c, ~, r be primitively recursive functions [3] such that for all x, N~N

lc(x, g)=x, rc(x, g)=gand c(l(x), r(x))=x.

We will now state basic definitions and formulate the studied questions.

Let ~=<B, V, A, C> be a Boolean algebra. A recursively enumerable numeration of the Boolean algebra ~ is a numeration v:N § B such that there exist recursive functions Iv, ]A,

fc such that for all x, y~N there holds

v] v (x, y) = vx V vg,

Vlv(X, y) =vx A vy,

Translated from Sibirskii Matematicheskii Zhurnal, Vol. 24, No. 6, pp. 36-43, November- December, 1983. Original article submitted September 2, 1981.

852 0037-4466/83/2406-0852507.50 �9 1984 Plenum Publishing Corporation

wfc(x) = Cv(x), and the equality relation N~{(x, y ) l v x = v y } is recursively enumerable. If qv is recursive, then the numeration v is called a constructivization of the Boolean algebra

If we have a recursively enumerable numeration w of a Boolean algebra ~, then we can define the Boolean algebra

~5~ ~ <N/q.,, V,,, A ~ a,>, where

xAl~V~y/q~ ~ Iv (x, y)/~l~, x l ~ A~, y/~l~ ~ ]A (x, y)/~l~,

C~(x/*l~) -~ ~(x) /~ .

The Boolean algebra so defined is said to be recursively enumerable (r.e.); if ~v is recursive, then such a Boolean algebra is said to be recursive.

It is easily seen that theBooleanalgebras ~9~ and ~ are isomorphic under the natural map (p~(x/~) ~- v(x).

If ~ and ~ are two recursively enumerable Boolean algebras, then a homomorphism q0:~-+~, is said to be recursive if there exists a recursive function f such that q0(x/N~)= /(x)l~l~.

It is readily seen that this exactly corresponds to recursiveness of the homomorphism q~ [1] of the enumerated Boolean algebras (@[, w) and (~, ~) where ~ -t , -~ ~ ~

We define the product of two recursively enumerable numerated algebras (~, v) and (~, ~I by putting (~X~, vX~), where ~[)<~ is the Cartesian product of Boolean algebras and (vX~)(c(x~ y))~<w(x), ~(y)>. Obviously, we obtain a recursively enumerable Boolean algebra again. Thus, we can also define the product of two recursively enumerable Boolean algebras ~ and ~ by taking ~x~ ~ <TV/~x~, V,x~, A~x~, C~x~>.

It is clear that ~ X ~ x ~ In this case, we will denote the r.e. Boolean algebra ~x~ by ~X~ �9 Two recursively enumerable Boolean algebras ~f~ and ~ are said to be recursive~y isomorphic if there exists a recursive isomorphism q0:~v--~.

o n t o

It is obvious that in this case the isomorphism ~-~ is also recursive. If ~ and ~ are recursively isomorphic, we will denote this by ~[~-~.

If a class ~ of r.e. Boolean algebras is given, then, following Hanf, we call a r.e. Boolean algebra ~ universal in the class JE, if ~J~ and for each roe. Boolean algebra ~9~2{ there exists a r.e. Boolean algebra ~= such that ~X~-------~.

We consider the problem of existence of universal Boolean algebras for the following three classes of r.e. Boolean algebras:

B, atomless r.e. Boolean algebras,

A, atomic r.e. Boolean algebras,

Ar, atomic recursive Boolean algebras.

In all three cases, negative answers are obtained. In spirit, all these results are close to [5].

Proposition 1. If ~,, ~9~ , and ~= are r.e. Boolean algebras and ~[~-~X,~, then ~ is a recursive Boolean algebra if and only if ~9~ and ~ are recursive Boolean algebras.

The proof is obtained by direct verification using the recursive isomorphism. It is known that an atomless Boolean algebra has, up to an autoequivalence, only one constructivization [I], so any two recursive atomless Boolean algebras are recursively isomorphic~ So fix one of these constructivizations v for the atomless Boolean algebra B=~ and consider the recursive Boolean algebra By constructed for it.

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Proposition 2. For each r.e. Boolean algebra ~[~ there exists a recursively enumerable

congruence ~ on B w such that ~-BJ~].

Proof. We first construct a generating tree D for the r.e. Boolean algebra ~. Con-

sider an element no~N such that n0/~ = I and put g(0) = no. Suppose that we have already defined the value g(x) for all x~ U E~ We now define g on elements of Et+1. Let x~Et+i,

i~<t

then LP(x) = x or RP(x) = x, where the functions L, R, P and the sets E i are defined as above.

D e f i n e g(L(P(x))) ~- fA (g(P(x)), t), g(R(P(x))) ~ fA (g(P(x)), ]c(t)).

The obtained set D~{g(n)/~l~In~N} is a generating set in ~,~. Using the function g, one can easily define a recursive homomorphism of the recursive Boolean algebra ~ onto ~. It is readily seen that the kernel of this homomorphism is recursively enumerable. Taking now

the congruence ~ determined by this kernel, one can see that it is as required.

Thus, the study of r.e. Boolean algebras is reduced to the study of quotient algebras

of atomless Boolean algebras over recursively enumerable congruences.

We will prove another property of r.e. Boolean algebras.

It is known that for each a~ holds ~0=X~c(=~ [I]. We will obtain a recursive analogue of this statement.

Let (~, ~) be a r.e. numerated Boolean algebra and e~I~I In this case the set {xlw(x)

~a} is recursively enumerable. Consider an arbitrary recursive function such that f:N § onto

{xl~x~a}. The numeration v~(x)~v~(x) of the Boolean algebra ~= is, as can be easily seen, recursively enumerable. It is just as easy to see that this numeration does not depend, up

to a recursive isomorphism, on the choice of a function f. Thus, we have obtained a r.e. numerated Boolean algebra (~=, ~=) and a r.e. Boolean algebra (~a)Va"

Clearly, ~_~r(~a)~aXr(~c(a))~C(a) It is easy to prove the following converse statement.

Proposition 3. If ~, ~, ~= are r.e. Boolean algebras and ~ ~(~X~=), then there

exists an element a of the Boolean algebra ~[ such that ~--~-r(~[a)va.

Thus, a r.e. Boolean algebra ~ is a direct summand in a r.e. Boolean algebra ~, if and only if there exists a recursive isomorphism q9 onto (~a)va for some a, and this is equiv-

alent to the existence of a recursive function g such that r for all x~N.

Let q0~(x) be a universal p.r.f, for the class of p.r.f, in one variable [3], and let r be defined if q~(x) is computed in less than t steps, and r and undefined otherwise.

We now turn to the solution of the questions outlined in the beginning of the article.

THEOREM I. There is no universal r.e. Boolean algebra in the class B of atomless r.e. Boolean algebras.

Proof. Suppose, to the contrary, that there exists a universal r.e. Boolean algebra ~= in the class B. Let Ord=UOrd t, where Ordt=-Ord TM and {Ordtlt~N} is a strongly computable

sequence [3] of finite sets, and Ord~{<x, y>l~x~y} is a recursively enumerable set. We will now construct a r.e. Boolean algebra ~ in B which is not a direct summand in ~. Let B be a recursive Boolean algebra constructed for (~m w~) , where ~ is constructed for the func-

tion fN, fN(n) = n for all n~N.

Consider recursive functions e, h, g such that e(n) = k if w~(e)=k, wNg(n)=NL~n(o ) and h is

such that ~Nh(n) = An, and numbers n0, n~ for which wN(n0) = ~, ~N(nl) = A0. Henceforth, we will omit the subscript N in the proof and construction.

t In the course of construction, we will construct finite sets ~t, Un ' functions r(n, t)

and ~t.

Step 0. Put ~l~176 r(n, 0)=0 for all n~N and 50 is not defined anywhere.

Step t = c(n, m) + ~.

Case I. If ~t-l(n) is undefined but q0t~(g(n)) is defined, then define ~t(n)~g(n) and leave the rest unchanged, i.e., r(m, t)~r(m, t-- ~) and Utm~ Umt-~, ~it ~- ~-~ for all m.

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C a s e 2. If ~t-~ is defined, <q~.(g(n)), no>~Ord ~, {x~r(n, t--t) l<x, q~.(g(n))>~Ord~}~ {x I (~Ig~t)(x=~$(y) V (<x, ~(g)> ~Ord tV <~nt (y), x>~Ordt))}, q~$ preserves operations for elements

x such that x~r(n, t), wx<~vg(n) and x~p~n , i.e. for all x, y, z in this set there holds:

If uz = ~xAvy, then

t ~ t I~ (~n ~ (~))> <~ (z), (x), Ord/ t t t and <Ia(qo~(x), q)n(g)), q~.(z)> ~ Ord t,

t O r d t ) , and a l s o <x, n 0 > ~ O r d * w h e r e A ~ { V , ~ }, a n d ( E Y ~ ] ~ - ~ ) ( ( v x = O V v x ~ Vvhy~)=~@n(x), no>

f o r e a c h x ~ U.tn -~ ; t h e n p u t

~t(n) -~ hRe~'-~(n),

U t ~ T./.t--i ~ U lhCe(-~(n)}, ~]' ~ ~l '-~ U {hLe~'-Kn)},

r ( n , t ) ~ r ( n , t - - t ) + t , t h e r e s t i s u n c h a n g e d .

If the conditions of the first two cases are fulfilled, then we turn to the next step leaving everything unchanged.

We denote by I the recursively enumerable ideal generated by the recursive]y enumerated set N=UN ~ . Consider the recursively enumerable quotient algebra DN/I. It is readily seen that DN/I is isomorphic to the direct sum

~-J f~vg(n)/[ ~ [ ~vg(n)l" n ~ N

We w i l l show t h a t e a c h surmmand i s a n a t o m l e s s B o o l e a n a l g e b r a .

I t i s e a s i l y s e e n t h a t f o r e a c h n Case t c a n o c c u r o n l y o n c e . We w i l l show t h a t Case 2 , t o o , o c c u r s o n l y f i n i t e l y many t i m e s f o r e a c h n . S u p p o s e t h e c o n t r a r y . Then f o r a l l t <%g(n), n 0 > ~ O r d ~ a n d , t h e r e f o r e , ~ % g ( n ) r . C o n s i d e r two e l e m e n t s a , b s u c h t h a t 0 < a , b<r~cp~g(n) a n d a/? b = O �9 S u p p o s e t h a t a h a s ~ - n u m b e r n a a n d b h a s ~ - n u m b e r n b . S i n c e C a s e 2 occurs infinitely often, there exists a step t0 such that na, n b < r(n, to) and

{n~, nb} --~ {x [ <x, ~ng (n)> ~- OrdtO}.

In this case after the step to there exists a step tl when Case 2 holds for n. But then ^

there are elements n a and fib such that

q)n ~na) .-~ tlnaand q)n ~ nb } ~ tlr~b

[ h e r e , x ~ . t g ~ x = g V ( < x , g>, <g, x > ~ O r d t ) ] , w h e r e v n , ~ v g ( n ) a n d v n b ~ v g ( n ) . T h e r e f o r e , t h e r e

e x i s t f a m i l i e s no . . . . ,n k a n d mo . . . . . m l s u c h t h a t v n a = A n o U . . . U A ~ ~ a n d v~nb=Amo[J . . .UA,~

( s i n c e nn~v~O, ~n~4=O, ~n,f) nnb=O ) a n d a n u m b e r e s u c h t h a t

Ee =- {mo, . . . , rnt, no, . . . , nk}, n~ ~ e(g(n))

f o r i <~ k, ml ~-< eg(n) f o r i ~< l , and {no . . . . . nh} N {too . . . . . mz} = 25.

C o n s i d e r now t h e s t e p t * s u c h t h a t C a s e 2 h a s a l r e a d y o c c u r r e d e t i m e s f o r n ; t h e n a l l e l e m e n t s h ( x ) s u c h t h a t x ~ e g ( n ) a n d x E E e , e x c e p t , p e r h a p s , o n e , a r e c o n t a i n e d i n t h e i d e a l

t* generated by qt*, but in this case there exist l l , . . . , l p ~ v M such that

P p

~na < U vhli V vnb < U vh (Id, i = l 6=1

so a t t h e n e x t s t e p t2 > t * , when Case 2 w i l l o c c u r a g a i n f o r n , we w i l l h a v e <na, no> ~ Ord t~

o r <nb, no>~Ord % b u t t h i s i s i m p o s s i b l e b y t h e c h o i c e o f e l e m e n t s a ~ 0 a n d b ~ O.

S i n c e t h e i d e a l I i s g e n e r a t e d b y { v ( n ) f n E U N q a n d q t c o n t a i n s n u m b e r s o f e l e m e n t s h a v i n g n o n e m p t y i n t e r s e c t i o n w i t h u g ( n ) o n l y when C a s e 2 h o l d s f o r n , N = U q t c o n t a i n s o n l y

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finitely many elements having nonempty intersection with vg(n), and all of them are strictly

less than vg(n). Suppose they are a~f...,a n < vg(n). In this case ~vg(n)~---~Ua~X~vg(n)\ua~, and Ua~=vg(n) , thus

~3vg<n)/I (] 1 ~vg(n) I ~ ~vg(n) \ Ua i

Therefore, since ~vg(n)\Ua~ is an atomless Boolean algebra, DN/I is also an atomless Boolean algebra.

We will show that DN/I is not a direct summand in ~= . Suppose the contrary, then there exists a recursive function ff~ implementing an isomorphic map of DN/I onto (~)= for some a, i.e.

(Vxy) (n%/v(x , g) ~ ( x ) V ~%(y)& n%/A(x , y ) = n % ( x ) A a%(y)& (vxI = O - ~ n ~ ( x ) = 0))

and, if vx/l = 1, then

(vy ) (~y ~ ~%(x) ~ (~z) (~%(z) = ~y)).

Consider the step tl after which Cases I and 2 no longer hold for n. Then for all t tl r(n, t) = r(n, tl). Since ~ is defined everywhere, Case ] will necessarily hold for n;

therefore, ~t1(n) is defined. Since ~ implements an isomorphism on (~)= for some a and ~ is defined everywhere, there exists t2 = c(n, m) + 1 > tl such that at the step t2 all conditions of Case 2 are fulfilled for n, but then the construction is valid for n by Case 2, which is impossible. The obtained contradiction proves the theorem.

We will now prove similar results for A and Ar. Both statements will follow from the

following fundamental statement.

THEOREM 2. For each atomic recursively enumerable Boolean algebra (2, ~) there exists a recursive atomic Boolean algebra which is not a direct summand in (~, ~).

Proof. Like in the previous theorem, we consider the Boolean algebra (~N, vN) and partial recursive functions g, h, and e such that ~g(n)=ALRn(o ), vh (n )=A~ for all n, and e(n) =

k if and only if ~n = Ak. We will define a subalgebra ~ in ~N by steps. At the step t, we construct the finite Bt~N and define the value of the function r(n, t) for all n.

This construction combines the ideas of proofs of the previous theorems.

Step 0. B~ {v-'(~), v-t(A0), v-~(A~), v-~(Az)} and r(n, 0) = 0 for all n ~ N.

Step t + I. Let t = c(n, 1). If the following conditions are satisfied for n:

(I) ~ is defined on g(n),

(2) ~$ is defined on Bt~{i[i~r(n, t)},

trBt~ { i l i ~ r & Tng(n)>~ (3) qnL ) ~__ (n, t) <i, Ord~},

(4) ~ preserves operations for elements x such that x ~ r(n, t) and x ~ B t, i.e., for

all x, y, z~Bt~{~li~r(n, t)}, if ~xhwy = uz, then

<~ (z), /~ (~n (x), ~n (9))>, </~ (% (x), % (y)), % (z)> ~ Ord t,

where A~{V, A} �9 if vx ~ O, then <T~(x), n0>~Ord t, if wx = O, then <~(x), n0>~Ord t , if vx

vy, then

(5) for each

<%(x), %(y)> ~ Ord t V <%(y), %(x)> ~ Ord t,

x ~ {x ~ r(n, t) l<x, %g(x)> ~ Ordq

t h e r e e x i s t s z ~ B t such t h a t <%(z), x > ~ O r d t and <x, % ( z ) > ~ O r d t,

t h e n i n t h i s c a s e we d e f i n e r(n, t - F t ) ~ r ( n , t ) + l , r(m, t T l ) ~ r ( m , t) f o r a l l m # n and

B T M ~ v- i (gr (vB ~ U {vg(t+ l)} U {~;h(k)tk ~ g(n) and k ~ E~(~, o+l}))-

856

If the listed conditions are not fulfilled, then, taking r(m,~+i) ~r(m,t ) for all m and

B '+* ~ v-*(gr (vB' O {vg(t + ~)))),

we turn to the next step.

As before, in our construction Ord is the recursively enumerable set of pairs <x~ y>

such that ~x ~< ~y and Ordt, t ~> O, is a strictly computable sequence of finite sets such that UOrd~=Ord and Ordt+l--~Ord t for all t, and also ~n0 = 0.

Let ~Ir with the underlying set B, where B~0/~ ~, In '~ , for each a~x there

e x i s t n l . . . . , nk s u c h t h a t a~ g( n i ) V...Vvg(n~) o r c(a)~vg(n~)V...Vvg(n~). So ~ = : ~ ( l ~ } [ D ~ g ( n ) ) .

I n o r d e r t o show t h a t ~ i s an a t o m i c B o o l e a n a l g e b r a , i t s u f f i c e s t o n o t e t h a t f o r e a c h n t h e a l g e b r a [ ~ [ n ~ e ( ~ ~ i s f i n i t e . S i n c e a t e a c h s t e p c ( n , 7) + 1 o n l y t h e e l e m e n t ~ g ( n ) and s m a l l e r e l e m e n t s c a n be a d d e d t o ~ , e l e m e n t s w i l l b e a dde d i n ~g(=~ o n l y a t t h e s t e p s c(n, l) + I if all five conditions of this step are fulfilled. We will show that all five conditions for n can be fulfilled only finitely many times. Suppose the contrary. But then ]im

t-+oo

r(n, t) = oo and so all elements of ~ less than ~g(n) will eventually be added to ~ , but in this case the element ~g(n) is atomless in ~ . Since ]imr(e,t)=oo and all five cases

t~oo

for n occur infinitely often, q~(k) is defined for all k such that vk~vg(e) . Define a map for all vk such that vk~wg(e) putting ~(vk)-~(k ~) , where k' is the smallest such that

~k' = vk. It follows from the definition that ~ is well-defined, and condition (4) for steps c(n, ~) + I which holds infinitely often implies that ~ preserves the operations A and V and ~(z)---~0~=~x=0, i.e. ~ is an isomorphic embedding. Condition (3) for the same steps implies that ~ is an isomorphism onto [~[~ng(m[ but in this case @ maps an atomless Boolean

algebra isomorphically onto a direct summand of ~ , which contradicts the atomic property of the Boolean algebra ~. Therefore, ~ is an atomic Boolean algebra. Consider a construc-

tivization ~ of the Boolean algebra ~, taking a recursive function f: ~ which exists O~IIO

since B is recursively enumerable and putting ~(n)~-~v](n). Since ~ is a constructivization, is obviously a constructivization of the subalgebra ~ of the algebra ~-.

It remains to show that (~, N) is not a direct summand in (~[, ~). Suppose otherwise~ then there exists a recursive function ~ such that ~-~ is an isomorphism of the Boolean algebra ~ onto

Consider a function q~ such that r . Then the map ~q%v -~ is also an

isomorphism of the Boolean algebra ~ onto ~[u~a-~(~) and, therefore, }t~pnv-~} (Z)vg(n) is an iso-

morphism of (~O)~e~ onto ~lu~ng(n ) . But in this case consider the step to after which conditions

('I)-(5) are not fulfilled for n at the steps c(n, 7) + ~. Then, after the step to the value

of r(n, t) does not change; since >q~nv-~ } (~)vg(n is an isomorphism of (~)~g(~ onto g~ng(n>

there exists tl > to such that all conditions (I)-(5) already hold, but then, taking 7 such

that c(n, 7) + I ~> tx, we obtain that all conditions (~)-(5) are fulfilled for n, which con-- tradicts the assumption. The theorem is proved.

COROLLARY I. The class Ar has no universal recursive Boolean algebra.

COROLLARY 2. The class A has no universal Yecursively enumerable Boolean algebra.

Note that the present article is a continuation and development of ideas started in [5-7].

We n o t e , i n c o n c l u s i o n , t h a t q u e s t i o n s o~ e x i s t e n c e o f u n i v e r s a l a l g e b r a s i n t h e c l a s s e s A, A r , a n d B became known to t h e a u t h o r f r o m P r o f e s s o r W. Hanf who u s e d u n i v e r s a l r e c u r s i v e l y e n u m e r a b l e B o o l e a n a l g e b r a s f o r a c h a r a c t e r i z a t i o n o f r e c u r s i v e l y a x i o m a t i z a b l e t h e o r i e s [ 8 ] .

I ~

2. 3.

LITERATURE CITED

Yu. L. Ershov, Theory of Numerations [in Russian], 3~ Novosibirsk Oniv, Press (1974). R. Sikorski, Boolean Algebras [Russian translation], Mir, Moscow (1970). H. Rogers, The Theory of Recursive Functions and Effective Computability [Russian translation], Mir, Moscow (1972).

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.

5.

6.

7.

8.

M. G. Peretyat'kin, "Strongly constructive models and numerations of the Boolean algebra of recursive sets," Algebra Logika, 10, No. 5, 535-557 (]971). S. S. Goncharov, "Some properties of eonstructivizations of Boolean algebras," Sibo Mat. Zh., 16, No. 2, 264-278 (1975). S. S. Goncharov, "Nonautoequivalent constructivizations of atomic Boolean algebras," Mat. Zametki, 19, No. 6, 853-858 (1976). A. T. Nurtazin, "Computable classes and algebraic criteria of autostability," CandidateVs Dissertation, Mathematics Institute, Siberian Branch, Academy of Sciences of the USSR, Novosibirsk (1974). W. Hanf, "Characterization of B<2>," Preprint, University of Hawaii, Honolulu (1979).

PICARD'S THEOREM AND HYPERBOLICITY

M. G. Zaidenberg UDC 517.5

INTRODUCTION

In this paper we investigate the interrelations between the multidimensional analogs of the small and big Picard theorems and Montel's theorem. Generalizations of these classical theorems have been obtained in the last 10-15 years in the framework of hyperbolic complex analysis (see []-3]). Let us recall the necessary statements.

The Small Picard Theorem. Every holomorphic map J:C-+CP~\{0; i; ~} of the complex line into the Riemann sphere with three punctures is constant.

The Big Picard Theorem. Every holomorphic map J:&\{0}-+CPI\{0; i; ~} of a punctured disk into a sphere with three punctures can be extended to a holomorphic map f:A § CP l of the disk into the Riemann sphere.

Montel's Theorem. The space of holomorphic maps Hot(A, CPI\{0; i; ~}) of a disk into a thrice punctured sphere is relatively compact in the space Hol (A, CP I) of holomorphic maps of a disk into a sphere (both spaces are endowed with compact-open topologies).

In generalizations the pair (CP i, CP~\{0; I; ~) is replaced by a pair (M, Y), where Y is a submanifold (as a rule, a domain) of a complex manifold M. The connection between these theorems and hyperbolicity in the sense of Kobayshi is as follows: for a hyperbolic manifold the small Picard theorem is true; for a pair (M, Y), where u the big Picard theorem is satisfied if Y is hyperbolically embedded in M; the latter condition is equivalent to the truth of Montel's theorem for the pair (M, Y) (see [2, Theorems 3.5, 3.6, and 8.4]). The property of complete hyperbolicity (see [2, Theorems 8.3 and 3.11]) works in other generalizations.

It turns out that if Y = M is a compact manifold, then all three theorems are equivalent. This follows from the results of Brody [4] by virtue of the statements made above, and it is generally speaking false in the noncompact case [I, p. 92], [6]. Green [5] con- sidered a domain Y = M\ D, where D is a closed hypersurface in a compact manifold M. In this situation the truth of the big Picard theorem and Montel's theorem for the pair (M, Y) follows from the truth of the small Picard theorem for the submanifolds Y and D (more precisely, for Y and all the "bounding strata" of Y determined by the irreducible components of D).

We study the class of locally complete hyperbolic (l.c.h.) submanifolds, introduced in the papers of KobayashiandKiernan [7, 8]. It turns out that for l.c.h, submanifolds an analog of Green's theorem is true (Sec. 2, Theorem 2.1). The proof is carried out according to the scheme of Brody and Green. The property of being l.c.h, is satisfied by closed submanifolds of polyhedral domains (Sec. 3), inparticular~ affine manifolds, polynomial poly- hedra, and so on. In Sec. 4 we give the conditions for hyperbolicity and hyperbolic embeddedness of the inverse image of a hyperbolic manifold in terms of the fibers of the map. In particular, we obtain here a positive answer to Problem 10, posed by Kobayashi in [1], and we give a criterion for complete hyperbolicity and hyperbolic embeddedness in CP 2 of an

Translated from Sibirskii Matematicheskii Zhurnal, Vol. 24, No. 6, pp. 44-55, November- December, 1983. Original article submitted October 2, 1981.

858 0037-4466/83/2406-0858507.50 �9 1984 Plenum Publishing Corporation

universal recursively enumerable boolean algebras

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