Ehrenfeucht games and ordinal addition

ANNALS OF PURE AND APPLIED LOGIC ELS~IER

Annals of Pure and Applied Logic 89 (1997) 53-73

Ehrenfeucht

games and ordinal addition Franqoise Maurin

GRAL, Matht!matiques, UniversitP de Caen. 14 032 Caen Cedex, France

Abstract We show in this paper that the theory of ordinal addition of any fixed ordinal w’, with c( less than ow, admits a quantifier elimination. This in particular gives a new proof for the decidability result first established in 1965 by R. Biichi using transfinite automata. Our proof is based on the Ehrenfeucht games, and we show that quantifier elimination go through generalized power.

R&urn&On montre ici que, pour tout ordinal c1 inf&ieur d ow, la thCorie additive de o’ admet une Climination des quantificateurs. On obtient ainsi une nouvelle preuve de la d&cidabilitb de ces theories (rCsu1tat de R. Biichi de 1965, bask sur des automates transfinis). On utilise ici les jeux d’Ehrenfeucht, avec un formalisme B la Ferrante et Rackoff, et on montre que le fait d’admettre une Climination des quantificateurs se transmet par produit gtnCralisC.

0. Introduction Biichi proved

studying

in [ 1, 21, that the theory

Bi.ichi’s decision procedure,

of addition

of every

ordinal

we find [l l] an exponential

is decidable.

When

tower of height O(n).

A result of Meyer [13], improved by Compton and Henson [3], says that the weak second-order theory of (FJ,S) (where S is the successor function) is not decidable in time less than an exponential tower of height dn, for some d. It is not hard to prove that the weak second-order theory of (a, < ) is reducible in polynomial time to the theory of (ma, +). Thus, the theory of ordinal addition, for any ordinal 02, with CI2 w cannot be decided in time less than an exponential tower of height dn, for a certain constant d. So Biichi’s algorithm is nearly optimal, even from a theoretical point of view. Since it is rather convoluted technically, it appears a very natural task to try to find more simple alternative algorithms, such as the ones provided by the point of view of Ehrenfeucht’s games. We shall consider the partial order relation <, defined on finite sets of ordinals, that corresponds to the order on the singletons (we see easily that the weak second-order theory of (~1,< ) and the first-order theory of (‘@(a), c, 4~) are reducible one to each other). The notion of generalized power enables us to formalize the relations between the structures (We, +) and (‘p,(a), C, <) (see [lo]). Namely, for every ordinal CI, the 016%0072/97/$17.00 @ 1997 Elsevier PII SO168-0072(97)00005-5

Science B.V. All rights reserved

E: Maurin I Annals of Pure and Applied Logic 89 (1997) 53-73

54

structure (o”, +) is a generalized and Rackoff’s method, If two structures

power of (w, +) by (‘!&(a), C, -CC). In using Ferrante

we prove that admit a quant$er

of those structures

elimination,

also admits a quantijier

Frai’sse first showed that the problem

then any generalized

power

elimination.

of elementary

equivalence

between two structures

was equivalent to the existence of “p-isomorphism” (see [8, 9]), and this was translated in terms of games by Ehrenfeucht [4, 51. We introduce a formalism that is very similar to the one used in the Ferrante and Rackoff method, and we prove that the elementary equivalence of two structures is equivalent to the existence of relations defined on both of the domains that satisfy particular conditions. With this formalism, very similar arguments allow one to prove that a structure is bounded (in the sense of Ferrante and Rackoff in [7], remind in Section 2), and to prove that two structures are elementary equivalent. Thus, the proof of the result “boundedness is preserved under generalized power” can be slightly modified to obtain a new proof of the fact that “elementary equivalence is preserved under generalized power”, already proved by Feferman and Vaught in [lo] (they used transformation of formulas). We can go further in the “generalized power” approach, and we see that For every ordinal 2
is a generalized

power of (Pf(co),

such that the structure

C, <> by (r(,(j),

c,<).

Thus, the boundedness of the structure (!&-(a), c,<) yields the boundedness of (!@(A), C, <), for every ordinal 1 < o.P (by a transfinite recurrence). Using the Ferrante and Rackoff method, we prove that the structure (‘$,(w), C, <) is bounded, and this shows that For every ordinal a < cow, the addition The paper is organized ness of a structure, we introduce

as follows.

Section

and a new formalism

the notion

of wa admits a quantijier

of generalized

1 contains

the definition

for the Ehrenfeucht power

[lo],

elimination. of the bounded-

games.

and we prove

In Section

2,

that bounded-

ness and elementary equivalence are preserved under generalized power. In Section 3, we show that ordinal addition is bounded, for every ordinal aa such that cL
1. The boundedness of a structure Our formalism will be similar to the one developed by Ferrante and Rackoff in [7]. Recall that the depth of a formula relative to some signature is defined by recurrence. The depth of atomic formulas is 0 and, for any positive integer n, the depth of any linear combination of formulas of depth n is also n, and the depth of (Vx)F(x,. . .) is n + 1, when the depth of formula F is n.

F. Maurin I Annals of Pure and Applied Logic 89 (1997)

Definition. lations

Consider

(denoted

equivalence

a structure

by (=fkkn,k))

relation

d.

of &‘, the family

such that, for n>O and k > 1, the relation

on (Dom &)k

a’ and b satisfy the same formulas to the signature

We call basic relations

55

53- 73

of re-

=;ei, is the

such that a’ =; ’ k b’ holds if and only if the k-tuples with k free variables

and depth at most n, relative

of .d.

Ferrante and Rackoff have proved that a sufficient condition for d to admit a quantifier elimination is that the relations =idk satisfy some bounded extension property, namely that, for each k-tuples an element b satisfying

and the element

ii, b’ that satisfy a’ =f+, k b’ and for each a, there exists

b can be chosen in some finite subset of Dom ~2 (depending

on a’, 6,

b,n,k). Definition. (i) A norm on d is any mapping N of Dom .c9 to w such that, for any n, the set {a E Dom &‘; N(a) < n} is finite. The norm of a tuple (XI,. ,xk) is the supremum of the norms of x1,. . ,xk. (ii) Let E = (En,k),,>0,k2 1 be a family of equivalence relations, such that the domain of En.k is (Dom ~2’)~. Assume that H is a function from FU3 to N. The H-extendible if and only if, for any naturals n, k and M (k # 0), for u’ and b’ of the domain satisfying a’E,+l,k&, if the norm of b’ is less than any element Lik+l of the domain, there exists an element bk+l that satisfies (aak+l)En.k+l(~,bk+,),

family E is any k-tuples M, then for the relation

and has norm at most H(n,k,M).

Proposition 1.1 (Ferrante and Rackoff [7]). Assume that the structure .d involves u jinite and relational signature. If the jbmily of basic relations of & is H-extendible (for

some jimction

H and some norm N), then .d admits a yuuntij5er elimination.

Actually, it is not necessary to use the relations =fk themselves, but one can use instead any convenient refinement. This leads to the following criterion. Proposition 1.2 (Ferrante and Rackoff [7]). Let ,rB be any structure that involves II ,jinite and relational signature. Assume that there exists a family of equivalence relations E = (l&k) that is H-extendible (for some function H and some norm N ), and such that EO,k refines =tk, for every k. Then structure .r4 admits a quantijier elimination. According to the approach developed in the late 50s by Fraisst [8, 91 (“local isomorphism”) and by Ehrenfeucht [4, 51 (“Ehrenfeucht games”), one can also use the basic relations or their approximations to compare two structures, and to establish isomorphism.

F. MaurinlAnnals

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Definition. Assume that JS?, a are structures for the same signature. (i) The basic mixed relations of d and a’, denoted by =fi9, are such that, for any a’ and b’ k-tuples

of Dom J&’ and Dom 98, respectively,

if a’ and b’ satisfy the same formulas

a’ =z~~ b’ holds if and only

with k free variables

and depth at most n.

that E = (En,k)n2~.k2 1 is a family of relations

(ii) Assume

such that the domain of

En,k is (Dom&)k

x (DomS3)k. - We say that E is non-trivial if for any n and k, for any k-tuple a’ of Dom d, there exists a k-tuple b’ of Dom 69 such that a’E,,kb holds, and for any k-tuple b’ of Dom93, there exists a k-tuple a’ of Dom& such that aE,,kb holds. - We say that E is extendible if for any n and k, for any k-tuples a’ and b’ of Dom d and Dom 58, respectively, that satisfy &+t,k, for every element ak+t in the domain of zf!, there exists an element bk+l in the domain of B such that (&ak+l)&(& bk+l) holds, and for every bk+l of Domg, we can find &+I in the domain of J&’ that satisfies the same condition.

Remark. When we deal with the boundedness of a structure, the non-triviality condition is included in the condition “family E is a family of equivalence relations” (because of the reflexivity). Actually, one could replace, in Proposition 2, the condition “relations E,,k are equivalence relations” by “family E is non-trivial”. However, in practice, the relations E,,k that one exhibit to show the boundedness of a structure are equivalence relations. The comparison criterion the following criterion.

for two structures

obtained

by Ehrenfeucht

in [5] leads to

Proposition 1.3. Assume that d and ~3 are two structures for the same (finite and relational) signature. Then the following are equivalent: (i) d and ~49are elementary equivalent; (ii) the family of basic mixed relations =,,d39 k is non-trivial and extendible. (iii) there exists a family of relations E such that EO,k refines =$” for every k, and E is non-trivial and extendible.

Proof. (i) =S (ii): Assume that the theories of the structures

& and 99 are equal. We and show that this family is non-

shall call =,,k the basic mixed relations of (&,W), trivial and extendible. Non-triviality: Consider the natural numbers n and k, and a k-tuple a’ in Dom &‘. Let 0 (respectively Y) be the set that contains all the formulas with k free variables and depth at most n, which are true (resp. false) for (&,a’), and let us consider the formula

m,...,

xk):

A fE@

f(xl,...Jk)A

A fGY

7fh,...,Xk+l).

F Maurinl Annals of Pure and Applied Logic 89 (1997)

53-73

51

The sentence (3 xi ) . . . (3 xk)(F(xI , . . . , Xk)) is satisfied in the structure ~2, therefore it is also satisfied in g, because of the elementary equivalence of sa2 and 93. Thus, there exists a k-tuple (bi, . . , bk) of Dom 93 that satisfies F. The k-tuples same formulas

with k free variables

We show in a similar way that, for every k-tuple in the domain

of DomS?,

a’xn,k h’ holds.

there exists a k-tuple

of ~2 that satisfies a’=,,k 6.

Extendibility: We consider natural numbers n and k (k # the domain of .d and ?J, respectively, that satisfy =n+i,k, Dom &. Like previously, let @ be the set of formulas with depth at most n that are satisfied by (2, ak+l), and Y be the satisfied by (a’,ak+t). A

a’ and b’ satisfy the

and depth at most n: the relation

Consider

.f(Xl>...,Xk+l)A

fE@

/j

the formula F(xl,.

,xk+l)

0), k-tuples a’ and b’ in and an element ak+l of k + 1 free variables and set of those that are not

defined by

7f(xl,...,xk+l)

J-EY

(the formula F is a characterization of the (k + 1)-tuples of Dom 93 connected to (&&+I) by the relation = ,,$+I). The formula G(xi,. . ,Xk) : (hk+l)(F(x~ , . . ,xk+, )) has k free variables, and its depth is n + 1. The structure & satisfies G(Z), so the structure a satisfies also G(6) (because we assumed the relation ~?=~+i,k g). Thus, we can find an element bk+l in the domain of 23 such that the formula F( bl , . . . , bk+l) is satisfied by the structure 2. The tuples (a’, ak+l) and (g, bk+l) verify the same formulas of depth at most n. The relation (%ak+l>

=n,k+~(&bk+~)

holds.

Because of the symmetry in a’ and 6, the family The implication (ii) =+ (iii) is trivial. (iii) =S (ii): To establish this implication, we Ehrenfeucht games [4, 51. Assume we are given signature. We suppose that the domain of d and

( = n,k) is extendible. shall briefly expose the principle of two structures d and 39 for the same a are distinct (otherwise we consider

isomorphic structures). Let n be any fixed natural number. We play a game G,(d,S?) with two players, called “Player I” and “Player II”. The game consists of n moves, and each move i is a pair (xi, yi), where xi is the choice of Player I, belonging to Dom .d or Dom S?, and yi is the choice of Player II, belonging

to the set that has not been

chosen by Player I. We shall denote by ai the element belonging to Dom& played at move i, and bi the other one (in Domg). Player II wins the game if the tuples (al,. . . , a,,) and (bl,...,b,) satisfy the same quantifier-free formulas. Ehrenfeucht proved in [5] that the structures d and @ (for the same finite relational signature) are elementary equivalent if and only if for every natural n, Player II has a winning strategy in the game Gn(&‘,39). Assume that there exists a family E = (E,,k ) (such that the domain of E,,.k is (Dom .QZ)~ x (Dom a)k) non-trivial and extendible, such that Eo,~ refines = O,k for every k. The following procedure gives an effective winning strategy for Player II in the game G,(.d,.93) (for every n). _ For every first move x1 of Player I, Player II can choose an element yi such that the relation xi E,_I, lyl holds, because E is non-trivial.

58

F MaurinlAnnals

- Suppose

the players

ai,. . .,ai,bl,.

(al,.

.

have

of Pure and Applied Logic 89 (1997)

already

made

i moves

53-73

(i > 0), and chosen

elements

. .,bi that satisfy the relation

.,Ui)En_i,i(bl,.

If Player I proposes

. . ,bi).

(1)

a,+1 in the domain of &, then Player II can find an element

bk+l

such that the relation

(a~,...,ai,ui+l)En-i-l,i+l(bl,. . .,bi,bi+l)

(2)

holds, because of the extendibility applied to (1). In the same way, if Player I plays bi+l in the domain of g, then Player II can propose ai+i in Dom d that satisfies relation (2). At the end of the n moves, the elements chosen by both of the players certainly verify b,). Because relation _EQ refines =o+, this relation implies that (al,. ..,a,)Eo,.(bi,..., the tuples a’ and b’ satisfy the same quantifier-free formulas, and Player II wins the game. 0 In Section 3, we shall need the following Lemma 1.4. The family

property.

of basic relation of any structure

Proof. The proof is similar to the one (in Proposition of basic mixed relation is extendible”. 0

2. The generalized Let us consider

is extendible.

1.3) of the result “the family

power a structure

d = (A,CA,e),

where e is a particular

element

of the

domain A. Moreover, we consider a structure 99 = (Qtr(B), ZB)), where p,(B) denotes the set of all finite subsets of B, and where ZB contains the relation of inclusion. Subsequently, the set of symbols (signature) and the set of their interpretation in the structure will be confused. We denote by A cB) the set of all mappings of B to A that map all elements of B but a finite number to the distinguished element e of A. As usual, the support of a function f is the set of all elements whose image is not e. If 1 is a tuple (f,, . . . , f~) of functions, we shall denote by I(X) the tuple (f,(x), . . , fi(x)). Definition. A relation Y on A@) is describable f of A@), we have the equivalence r(J‘)@@/=G(Ti,...,Tl), - G is a ZB-formula;

where

in (&,a)

if and only if for every tuple

59

F. Maurin I Annals of Pure and Applied Logic 89 (I 997) 53-73

~ for every i between

1 and 1, Fi is a CA-formulas

set {x E B; B + Fi(y(x))}. A structure

’

(A@), C) is a generalized

of C is describable

power of d power

is an extension

and Vaught have proved in [lo] that elementary

are preserved the notion

by .% if and only if every relation

in (&,a).

It is easy to see that the generalized Feferman

not satisfied by e’, and Ti is the

under generalized

of the weak power.

equivalence

and decidability

power. We shall now prove a similar result relative to

of boundedness:

Theorem 2.1. IJ‘ the structures

.d and 8 are bounded cfor some norms),

norm I/ 11~on ‘$3,(B) satisfy the triungle inequality every generalized power of d by :“Ais bounded.

(&RJYlle<\lXl\~

and if the

+ IIYI\B),

then

Proof. We consider two bounded structures .d and 24 (the structure B is (p,(B), C,), where the set CB contains c), and a structure % = (AcB), Cc), where every relation in Yc is describable in (&,B). We call as usual =E,k the basic relations of %. To prove the boundedness of V, we have to define a norm on A(*), a function H from N3 to N, and a family of equivalence relations on A 1~ that is H-extendible and such that I!& refines = 0,~. The structure .r4 (respectively 69) is supposed to be bounded so, by definition of the boundedness, there exists a norm /I (IA on A (resp. I/ 11~on ‘$3,-(B)), and a function HA (resp. HB), such that the family of basic relations, denoted =$ (resp. =tk) is HA-extendible

(resp. HB-extendible).

From the norms ]I llA and /) ]IB, one can define an application II l/c from A(‘) to N, such that \\J’llc is the supremum between - the norm // ])B of f’s support; ~- the supremum of the norms of J‘(x) (in the sense of 11 ]IA), for every member x’ of B. One easily shows that, for every M, only a finite number of function f satisfy l/J‘llc GM. Thus, 11I(c is a norm, and it is said to be induced by norms // \\A and I( 118. For every positive

integers

n and k (k # 0), let L(,n, k) be the number

of classes of

the equivalence relation =$ (L(n,k) is simply denoted L if this is not ambiguous). of the The sequence ~1,. , rL is a fixed enumeration of the different representatives -. -,“k classes. We shall allways assume ~1 = ek. For every subscript p between be the set {U E B; z(u) =fk r,}.

1 and L, and for every k-tuple z of ACB), let O,,(:) The set {Or(z), , O,(t)} is clearly a partition of

B, and the classes O*(z), . , @L(f) are finite subsets of B. Subsequently, precise the positive integers n and k involved in these notations.

we shall just

’ One easily sees that if the formula F, is satisfied by Z, then the set T, is infinite, and does not belong to the domain of .d.

60

F. MaurinlAnnals of Pure and Applied Logic 89 (1997) 53-73

For each relation

r of Zc, there exists G, a CB-formula

we can have several such formulas, us denote by di (respectively number

of free variables)

but for each relation

Y (of course

d2) a positive integer that majorates the depth (resp. the

of the formulas

Cc. We consider the function

that describes

Y, we fix one of them). Let

G,, when Y takes every possible

value in

W defined by W(0, k) = sup(3, dl ) + d2, and (for every n

and k),

We say that ]E,,Jj

holds if and only if the relation

is true (for every n and k, for all k-tuples 7 and J of (A(B))k). Since the basic relations =w and =& are both equivalence relations, to see that the relations En,k are equivalence relations as well.

it is not hard

Lemma 2.2. For every positive integer k, the relation EO,k refines =zk. Proof. We fix a positive integer k > 0, and two k-tuples 7 and g of A@), that satisfy the relation 7 EO,k g. We take the previous notations for n =0 and k. To show that every quantifier free Cc-formula is satisfied simultaneously by J‘ and i, it is sufficient to prove it for every atomic formula. We consider the formula R(v,, , . . . , v,,, ), where R is a relation of Cc, of arity a. There exists a formula G(Xi,. . .,Xl) of type CB, and I formulas F 1,. . . , Fl of type CA with a free variables, not satisfied by Z, such that R(3) is true if and only if the structure %? satisfy G(Ti, . . . , T,), where T, is the set {x ER d +:F;:(5(x))). S’mce the arity a is not necessarily k, we have to consider the formulas Hi such that Hi (VI,. . . , ok) is fi (v,, , . . . , v,,~). For every i between 1 and 1, we denote by Q the set {x E B; AS’bHi(7(X))}, and by Ui the same set relative to g (Q’ is {x E B; ~4 bHi(g(
all the subscripts

~1 such that H;(r,) is satisfied (the formulas Hi are not

verified by 2, SO none of the Ji’s contains 1). It is clear that Ui is the set UPCJ, B,(i), and Vi’ is UPEJ, O&j). The following claim says that if tuples are connected by some basic mixed relation, then one can consider union of those sets (the same union on both sides), and assert that they satisfy another basic mixed relation (with decreasing depth). Claim 2.2.1.

Assume that Z1, . . . , 2, and Zi, . . . , ZA are jinite and disjoint subsets of

B, that satisfy (Zl,...,Zm)

=:+*,m

F. Maurinl Annals of Pure and Applied Logic 89 (1997)

(with d > 3). Assume furthermore

Proof.

53-73

61

that I,,. . . , I, are subsets of { 1,. . . , m}. The relation

One can easily find a c-formula

UNION, (Xi,. . . ,X,,, Y) (with depth 3), that is

satisfied if and only if the set Y is the disjoint union of Xl,. . .,X,, (for every u). Consider any Cs-formula H with p free variables and depth at most d, and let H*(Y,,...,Y,) be

(3x1) . . . (-&)(UNONs, A .

Ya,,,,. . >Ya, A > s,

..I\UNIONs~(Ya..,,,...,Ya,.,,,,~,)AH(X~,...,&),

where ai, 1,. . . , ai,s, is an exhaustive p + sup(3, d). The equivalences

enumeration

of Ii, for every i. The depth of H’ is

below hold.

%98+H*(Z,,...,Z,) (because

the depth of H* is d + p)

The tuples (U, E II Zi,. . . , Ui E Ip Zi) and satisfy the same Z:Bformulas with depth d, and the claim is proved. 0 The tuples (@(I), . . , &(j)) and (02(7), . . . , @L(I)) satisfy the relation = d,+sup(3,dz),~_i (this is our hypothesis j&kg). The previous claim enables us to affirm that the tuples ( UI,. . . , Ul) and (U,l, . . . , Ul) satisfy =$,/ (because 1 < dz). Then the following equivalences arise: +K?)

HB~GG(U~,...,UI) w &?k G( Ui,. . . , U,!)

because

the depth of G is at most dl

@ q k I%), and the refinement

property

is proved. Lemma 2.2 follows.

Lemma 2.3. The family E is H-extendible

0

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F. Maurinf Annals of Pure and Applied Logic 89 (1997)

53-73

Proof. Let us consider natural numbers n, k and M (k # 0), and two k-tuples 7 and $ of kB) such that the norm of 4 is bounded by M, and such that the relation (@z(7) 2* * * 9@L(h) ;=&flJ),L_, holds (with the previous

notations

(@2(g’),

* *. 3 am)

for IZ+ 1 and k). We consider

(1)

an element

,fk+, of

A@), and we shall construct gk+t in A @) that satisfies (7, fk+t ),!&+I (& gk+l ), and has norm below H(n,/c,M). To this end, we first construct a finite subset V of B, that “corresponds” to the restriction of the support of fk+t to the set @l(j), whose norm is bounded above by a certain function (that depends on HB). Next, we partition the sets V, @z(i),. . . , &-(d) with respect to the classes of the equivalence relation ==fk+, . These partitions enable us to define for every element u of B, some element a of A whose norm is below H~(n,k,A4), and that satisfies, if gk+t is this mapping, the relation (7, I-k+1 b%kflG>

f&+1

1.

Claim 2.3.1. We denote by U the support of fk+l. There exists a finite subset R with norm at most HB( W(n + 1, k), L - 1, k.M), that satisjies the relation (U

@2(7)

> * * ’ 9 W7))

=&+l.k)-1,L

(K

@2(Z),

. . ’ 3 @LG)).

V of

(2)

Proof. We apply the HB-extendibility of =,q to (1) and set U. For ,U= 1,. . . , L, the set O,(g) is included in the union of the supports of ~1,. . .,gk, so its norm is below k.M (because of the triangle inequality). We thus obtain the existence of the set V as said in the claim. 0 One constructs easily a c-formula of depth 2 that defines the relation “. . . are disjoint sets”. The tuple (U, @z(f), . . . , O,(J)) satisfies the latter relation, and by relation (2) so does the tuple (V, 8&J), . . . , @(ij)). We consider the equivalence relation =;ykfl. Let us denote equivalence classes of this relation, and by st = ek”, s2,. . . ,s~f sentatives of its different classes. The sequence sr, _. . , SL/ gives We shall imitate these partitions to define U OZCJ‘), . * * 3 @L(T). v,

@2(G),

by L’ the number of a sequence of reprepartitions for the sets partitions of the sets

* *. 3 @LG).

For every subscript

v between 1 and L’, let C,, (f, fkft ) be the set of all members u of B that verify (f,(u),. . . ,fk+&d)) =;$+I s,r. The set Cl (l,fk+t ) is infinite, but we shall consider only its restriction to the finite set U. For every v between 1 and L’, and p between 2 and L, let us introduce the (finite) subsets of B:

Claim 2.32. There exists a partition { Y,,l, . . . , Y,,LI } of V, and, for each subscript #u=2,. , . ,L, there exists a partition {Y@,J,.. . , YP,~l} of O,(i), that satisfy C&J

> -*,&,d

=&nf,.k)_l-~~‘,LJ,’

(~,1,-di.L’)-

(3)

F

Maurinl

Annals

of Purr

und Applied

Proof. We iterate the HB-extendibility lation

(2), and introducing

existence

Logic

63

53-73

=-‘, applying

of relation

successively

89 (1997)

it initially

to the re-

, XL,L~. We thus obtain

the sets X1.1..

the

of finite subsets of R, Yl.1,. . .,YLJ~. that satisfy .,@L~hJ,.,,.

(o:@,(f),. _.d -W(n-l

I.k)-I-

LL’.L.L’

(V.

.‘XL,I’) 02(3,.

> @L(i).

YI.1.. . .

Y/,.Lf

).

(4)

,u between 2 and L, the set {XIL.,,;I 6 v < C’} is a partition of 1< v
subscript

is a partition of V, and {YP,“; 1
Y/I =’$.k

(.f~(~),...,.fk(~),f‘k+~t~))

(cll(U~>.... Yk(~)). =Ztk+, .\v.

The family =,cJ is HA-extendible, so there exists an element J’ of A that has norm (in the sense of /( 11~) at most Ha(n, k,M), and verifies (.41(2:),...,.qk(~),y)

=fk+,

(.f;(u),.....lk(u),f;+l(uii,

we have @I(U),. . .,qk(u),y) =fk+, sl’, and we let gk+l(~) be y. The function gk+l we have defined so far belongs to A(‘), its norm is below H(n,k,M). It remains for us to show that the (k t I)-tuples (,f,,fk+1) and (G,cJ~~~) satisfy E,,+l. We denote by .&,(,$,cJ~~~) the set of all the elements of B that verify of gk+l, it iS Clear that, for every 1’ (gl(U)?...,%+l(u)) =$k+, s,. By COnStI-UCtiOn So

between 2 and L’, the set ,?I,,($ &+I ) is Uf,;, Y,,,,. (and C,.( .f,,fk+-+l) is the union of the same X,,V). Claim 2.2.1 (used to prove the “refinement property”) can be used here to prove that relation (3) implies the equivalence (~2(.j‘,.fkil),...,CLf(7,.fk+l))

=’

W’(,,k+,),[,‘-,

(~Z(~,,.qk+l)....,~L’(~.~k+l))

(by definition, W(n + 1. k) is w(n. k + 1) + U’ + L’). Family E is H-extendible, L.emma 2.3 holds. r7

and

64

F. MaurinlAnnals

Theorem

2.1 is proved:

H-extendible

and refines

As a corollary

of Pure and Applied Logic 89 (1997)

the family E is a family = o.k.

53-73

of equivalence

relations,

that is

0

to the proof above, we obtain a new proof of

Proposition 2.4. Consider structures d, a = (@t-(B), C,), 99’ = ($Jf(B’), ES), where the signature CS is jnite, relational, and contains c. Assume that V and 97’ are generalizedproducts of (&,93) and of (~2, SY), respectively, such that for each symbol of CC, its interpretation in V and its interpretation in 9?’ are describable by the same formulas. Then, if ~49and 23” are elementary equivalent, so are the structures +F? and $7’. Proof. We shall define relations on (A(B))k x (A(B’))k in a very similar way we did in the previous proposition. The basic relations of d and the basic mixed relations of (9’,93’) are, respectively, denoted by =& and =9@. For any natural numbers n and k, for any 7 and g (k-tuples of ,4(s) and A@‘), respectively), we say that TEn,k2j holds if and only if

is the set {x E B; c(x) =-01 n,k rP}, and O;(f) is the set {X E B’; r(x) ~1 = Z, r2,. . . , r-L is a fixed sequence of the representatives for classes of +$).

holds, where O,(f)

=,f$ rP} (the sequence the equivalence

The proof that the relation Theorem 2.1.

Eo,k refines =Tk”

IS exactly

the same as the one in

To prove the extendibility of family E, we have to slightly modify the proof of Theorem 2.1: each time we had “apply the HB-extendibility”, we have now to read “apply the extendibility”,

without

any norm condition

(the domains

we consider

here

We obtain finite subsets of B’, instead of finite subsets of B with norm at most . . . Actually, it is much more simpler, because we do not have to worry about bounding the norm of the element &+I that we construct. We just have to be sure that its support is a finite set of B’, and this holds, by construction. So we just have to prove that the family E is non-trivial, and this can be easily done: are not provided

Let us consider

with any norm).

natural numbers

n and

k (k #

0), and a k-tuple 7 of A@). We keep the

previous notations for II, k, and 7. We consider the (L - I)-tuple (O,(j), . . . , OL(~)) is non-trivial, so there exists a (L - I)-tuple (from 5&-(B)). The relation =$‘cf;,L_, (x, , . . . ,&) of !J$-(B’) that satisfies ’ (@2(j)

>...,

@L(7))

=g&,,L-*

GG >...> XL).

(5)

The sets O,(7) are disjoint, so the sets X2, . . . ,XL are either disjoint (because there exists a C-formula of depth 2 that defines the relation “. . . and . . . are disjoint sets”). We shall now define g. Consider a member v of B’. Just one of the two cases below is satisfied.

F MaurinlAnnals

oj’ Pure and Applied Logic 89 (1997) 53-73

Case 1: There exists a subscript

This subscript empty, element

is unique,

X, (because

For every i between

of @,(I).

2 and L) such that u is a member of X,,.

because the sets X2,. . .,X, are disjoint.

as its “counterpart”

is ambiguous,

p (between

65

of the relation

(5))

The set @,(T) is not and we consider

u an

1 and k, we let gi(v) be fi(U) (this definition

we can obtain several k-tuples

g of $$$(B’), but we just need to prove

at least one).

that we can construct

v does not belong to the set X2 U . U XL. Then we let gi( v) be e, for every i between 1 and k. We have defined functions gi, . . , gk from B’ to A. The support of gi is included in the (finite) set Xl U . . U XL. It is clear that, for every subscript p between 2 and L, the set X,, is Case 2: The element

{xE B’;

(gl(V), . . . ,gk(U))

=,$rp>>

so we have X, = O:(g). Relation (5) implies trivially jE,,kg. Because of the symmetry in B and B’, the family E is non-trivial is established.

2.4

0

3. Boundedness Definition.

and Proposition

of ordinal additicn (for ordinals below go”‘)

For any X and Y sets of ordinals,

X
means that sup XC sup Y.

We shall first prove that the structure (‘$$(o), C, <) is bounded (Proposition 3.1); then we shall see that, for every ordinal CI less than ww, there exists an ordinal Z less than CI, such that the structure (!&(a), C, <) is a generalized power of (‘$3,(o), C, <) by (‘$Jf(@, C, <) (Proposition 3.3). Then, a transfinite recurrence based upon those two results yields Proposition 3.4: “for all CI
3.5: “for all a < o”,

3.1. The structure

the structure

(5&-(o), C, K)

(09, +) is bounded’.

is hounded.

Proof. Let us introduce the predicates SINGL (“. . . contains a single element”), and VIDE (“. . . is an empty set”). Since they are both definable with inclusion, we can add them to the language . We denote by C* the signature {c, <,VIDE, SINGL}, and by =* the family of basic relations of the structure (‘$J N), C*). Recall that the family of basic relation of any structure is extendible (Lemma 4, Section 1). Let us consider the family of relations E such that il?,,kj is equivalent to A’ =&+, k B’. It is clear that for every n and k, the relation &,k is an equivalence relation, and Es,k refines =G,k. We shall prove that the family E is H-extendible, for some function H, and for the norm that associates with any finite subset of N its greatest element. Let us consider two natural numbers n and k (k # 0), and two k-tuples A’ and 2 of finite subsets of

66

F Muurin I Annuls of’ Pure and Applied Logic 89 (1997)

naturals,

53-73

that verify

A’ =?*+, k 3.

(I)

We consider Ak+t, a finite subset of naturals,

and we denote by NA its upper bound.

Lemma 3.2. There exists u function

N3 to N, and a positive

H(n, k,M

H jiom

integer s below

j, that satisjj

Let us assume this lemma for a while, and complete the proof of Proposition 1. Assume there exists a function H and a natural number s as in Lemma 3.2. We apply the extendibility of =* to the relation (II), introducing the set ,&+I. There exists a (finite) set Bk+l that satisfies (Al ,...>&{NA),Ak+l)

=;n+l,k+z (~l,...,~k,{~},~k+l)

(III)

It is easy to find a {c, <}-formula that defines the relation “the only element of X is the upper bound of I”‘. The latter relation is true for ({NA},Ak+l), thus it is true as well for the tuple ({s},Bk+i) (by (111)). Therefore, the upper bound of Bk+l is s, and it is bounded above by H(n, k,MB). Relation (III) implies trivially (AI,...>&+I) so

=;n+‘,k+i

(Blr...,Bk+l)>

family E is H-extendible.

0

We shall now mm to the proof of Lemma 3.2. Proof. By applying

extendibility

of =* to (1 ), we introduce

the set {NA}. Then there

exists a set S that satisfies

The set {N’} contains by s its element,

exactly one element,

by MA (respectively

so S is as well a singleton.

n/i,) the upper bound

Let us denote

of the set UF=, Ai (resp.

of u;=, Bi). _ The trivial case is when the inequality NA MA. Thus, we have also s >MB (because the previous relation is not satisfied by (Al,. . . ,Ak, {NA}), nor is it either by (Bl,. . . ,Bk, {s})). Definition. Assume that E is a X*-sentence in which the variables Xi,. .,&+I does not occur. The relutivized version of E is the formula we obtain when the sentence E

F Maurin I Annuls of’ Pure und Applied Logic 89 (1997)

is modified (VY)(.

61

53 -73

as follows: .) becomes

.),

(VY)((r? + Y A Y 4 X,+1 ) +

and (3Y)(. . .) becomes

(gY)(,?

< Y A Y + X,+1 17.. .),

where 2 + Y means r\f=, Xi + Y. (The free variables of ] sup d, sup &+ I[.) We denote by I?(g,Xk+l)

are restricted to be finite subsets

the relativized

A formula F(L?,Xk+l) is said to be relutivized equivalent to the relativised version of a sentence

version of a sentence E.

(with depth n) if and only if it is E with depth n.

Claim 3.2.1. Consider the naturul numbers n, k, M, u and v, and a k-tuplr 6 of jinitrsubsets of N,such that M is supo. qthe tuples (a,{u+M}) und(l.?,{v+M}) do not satisji =i,k+,, then there exists a relativized jtirmulu of depth n that is true jkw one tuple and false for the other one. We shall complete now the proof of Lemma 3.2, and postpone the proof of this claim (it seems quite natural, but is not so easy to prove). For every positive integers n and u, we denote by TH,(a) the set of all C*-sentences with depth at most n that are true in the structure (cP,(]O, a[), z‘* ). Let us consider (for every n) the set C$ of all C*-sentences with depth at most n. If we fix the names of the variables occurring in the sentences, it is a finite set. Each of the sets TH,(a) is a finite subset of G,, and there exist only a finite number of such subsets. Thus, for every natural number n, there exists D, in N such that the sets T@,(l), , TH,( D,) take all possible values for U&(a), when u is any natural number.2 The bounding function is defined by H(n, k, M) = M + DIRt:_ ], for every n, k and M. We consider a natural number s much greater than MB that satisfies the relation

(Al ,... >&{fYA)) =;“+L_,,k+,(BI>...>&>{Sl).

(11)

There exists a natural number z such that s is MB +z. By the definition of function D, there exists a natural number zl less than D2”~~__,such that TH2,,-~_,(z) is TH2rl+~_,(z, ), so the structures (r)f(]O,z[),C*) and (!$(]O,zl[),C*) verify the same sentences with depth 2n+2 - 1. For every C*-sentence

E, for every positive integers M and u, and for every k-tuple

,a of finite subsets of N such that sup 2 is M, the following

equivalences

hold:

Tf(10>4>J*) + E ++ ‘Pf(lMM + 4>,Z*> t= E, (because

(‘$Q(]M, M + a[), C* ) and (‘$Q(]O, a[), C* ) are isomorphic),

‘~(,(IM,M+~[),C*)~EE((~~~(~),C*)~=(~,{M+U}) (by definition

of the relativized

2 We should have say more formally T&(n) is TH,(y).

version

of a sentence 1.

that for every x, there exists an .v at most equal to D,, such that

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53-73

The tuples (8, {MB + z}) and (& {MB +zi}) thus verify the same relativized formulas with depth 2n+2 - 1. Claim 3.2.1 allows one to say that these tuples satisfy =;T”+Z_,,k+l. Let us consider SO= MB + zi . The relations (3, {so)) =&+2-i )k+i CR {ti

+ zl) =;n+LI,k+,

(A, {NAI)

hold, and SO is below MB + D,, which is H(~,~,MB).

Remark. We obtain the existence

of a function

Lemma 3.2 is proved.

H that bounds

the structure

0

(!Qf( N),

C, CC),

but we do not obtain an effective expression of such a function. Actually, one would like to prove that the number D,, is bounded by the number of .X*-sentences of depth at most f(n), for some function f. This fact, which seems natural, seems quite difficult to prove. Observe that, from a result of Meyer [ 131 (improved by Compton and Henson [3]), the function D, cannot be less than an exponential tower of height dn, for some constant d (we call exponential tower of height m the function EXPT 0 WER such that EXPTO WER(0) is 1, and EXPr 0 WER(t + 1) is 2ExpTowER(‘)). It remains

us to prove Claim 3.2.1.

Proof. Suppose that there exists a X*-formula @ which is true for (0, {U + M}) and not for (0, {u + A4)). Without loss of generality, we can assume that @ is prenex. Let @ be the formula

where the formula G is quantifier free. Step 1 (Tri-partition of the variables): The idea is that, for every positive integer x, “knowing a variable 5” is equivalent to “knowing the three sets 5 fl [O;M], $n]M; A4 +x[, and I$ fl [M +x; co[,‘. We shall convert every formula with ?, 2 and xk+i as fr?e variables into a formula with Q, 5,2, T,s,-? and &+i as free variables. If 2 and 2’ are tuples of finite subsets of N, we shall write V$‘Y...for

(V/r)@

4 Y 4.X!‘=?

Zi$‘Y...for

(3Y)(_? 4 Y +2/r\...)

. ..).

(these quantifiers can be read intuitively as “for every subset Y included in ] sup 2, inf_?[, we have . . .“, and “there exists a subset Y included in ] sup 2, inf g’[ such that . . .“. In a similar way, we define the quantifiers V=yY, V’=‘Y, which can be read as “for every subset Y included in [0, sup 81, we have . . .“, and “for _every subset Y included in [inf 2, co], we have . . .” (and we also define 3=~ and Fx). Let us consider the formula Q’:


Y1,2XQ1=xk+,

Y1,3

1..

.

CQm, xk+‘Ym,2)(Qm=xk+,r,,3)

F. MaurinlAnnals

of Pure and Applied Logic 89 (1997)

53-73

69

where G’ is the formula one obtains when replacing every occurrence of q by I;,1 U q.2 U Yj3 in G. The relation U is used here in a formal way, and will be eliminated subsequently. As sup I? is M, it is clear that the truth values of the formulas @‘( 0, {x+ are equivalent, for every x. in G’ each occurrence of the symbol

M}) and @(~,{x+M}) Let us now eliminate to the language. variables

Formula

X,&+1,

VIDE(q) Assume

G is a boolean

and of the following SINGL(I;)

we have the condition

combination

U , which does not belong

of atomic formulas

involving

the

formulas:

Y$+ q,

I; +4

Xl < q

qCXi

XiC$.

(that will be specified by the quantifiers)

(for every i). For each of the latter formulas, one can establish an equivalent formula with X, &+I, TJ, I;.,2 and IJJ as free variables. For instance, the formula VIDE(q,l U $2 U 5,3) is trivially equivalent to VIDE(I;,,) the formula (5~ U 5,2 U ?,3) 4 Xi is equivalent to ?,I +XiAVIDE(1;.,2)AVIDE(1;,3) {

$3

+

for i=l,..., for i=k+

Xk,l

and the formula (I;,1

U I;.,2 U 5,3)

I;,1 -x q/,, AVIDE&) VVIDE(q,,)

A

4 (${,I

U T/,2 U q,,3),

AVIDE(q,3)vVIDE(qr,,)

VIDE(I;J,~

A J$, +

A VIDE(I;,z)

A VIDE($,s),

k, 1, is equivalent A I;.,2 4

to

1;.‘,2 A

VIDE(q,3)

I;‘,3).

We replace in G’ every occurrence of those formulas (with variables q,~, 5,~~ T,J, and function U ) by the equivalent formula (with variables ?,I, I;.,2 and I;,J). We take the disjunctive normal form of the formula thus obtained. In each term of the disjunction, we put on the one hand the atomic formulas with 2,&+l as free variables, and on the other those with at least one Y as variable. If the subscripts 1 and I’ are different, variables Y.,,l and Y.,,a do not appear in the same atomic formula, so the formula so far obtained

(that we call fi) has the following

form:

(we denote by F’ the m-tuple (Y~,J,. . , Ym,/), where 1 is 1, 2 or 3). It is easy to see (when looking at the latter transformations) that in the formula F,,J, the variable &+I does not appear. Similarly, in FP,2, 2 and X k+l do not appear, and there is no .? in

F,,J. We denote by @I the formula

(Q;‘%,, ,(Q$+’ v pa

Fp,o(%&+d

YI,~)(Q~,+,Y~.~)...(Q,=~Y~,,)(Q~, AF,,,(~,f’)

A &2(y2)

a+’ Xn,2)(Qm,x,+, AFp,3(&+d3).

L.3)

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70

of Pure and Applied Logic 89 (1997) 53-73

At this point, it should be clear that, for every positive integer x, the tuple (0, {x+M}) satisfies @i if and only if it satisfies @. Step 2. Let Sz be the set that contains is true. For every subscript

all members p of Si such that F,,a(!?, { 1+M}) p, the formula F,,o is quantifier free. Thus, for every x # 0,

the truth values of F,,o( Q, {x +M}) C& the formula

(QT%J,(Q~;+~ l&P,Xj+1,

Yd(Ql++,

Tl, f2,

and of F,,o( i?, { 1 + M}) are equal. We denote by

Y~,,)...(Q~~ml,,,(Q,q+’

L,2)(Qm,+,

Ym,3)

f3),

where fi is V p,_S2F,,J(~, f’)

A F,,2(f2)

A &s(&+i,

f3).

It is not difficult to prove

that, for every x # 0, the tuple (0, {x + M}) satisfies 63, if and only if it satisfies @2. Step 3 (and last): Let di(X,&+i, Yl,i, . . . , Q) be the formula _

CQl$ r;+u )(Qj+l?xk+‘$+1,2)(Qj+1=xk+j

$+1,3)

. ..(Q~~r.,l)(Q~~+‘r,,2)(Q,,+,Y,,3) v

F,,1(~,~1)r\F,,2(~2)/\F~,3(xk+l,~3).

Pa

For every subscript j between m and 0, we denote by p(j) the property: “the formula Aj is a boolean combination of formulas GJ, G,,2, and Gfic,s, where - the formulas G,,, , Gp,2, and Gfi,3, have depth m - j; - the free variables of the formula G,,, are 2, Yl,i, . . . , F,,I, the free variables of G1,2 are among x, xk+l, YIJ,.. ., 5.2, and the free variables of G@,3 are xk+l, y1,3,.

. .,

53;

- the quantifiers in G,i are Q=‘, those in Glr,2 are @+’ (the variables 2 and &+i occur in Gfic,2only as bounds for the quantifiers), and quantifiers in Gp,s are Q=x,+, .” Property

9(m) is trivially

of property formulas

S(j)

true, and a decreasing

for any j between

recurrence

easily

shows the truth

m and 0 (we use the equivalence

between

the

and

WI MYI )v(VY,

)G’(YzW”r, P”(Y3 1,

and the dual equivalence). Property p(O) says that A0 (which is actually @2) is equivalent to a boolean combination (which we denote @3) of formulas G,,i, Gpc,2, and Gfl,3. Since the formula @3 can always be taken in disjunctive normal form, it is equivalent to

F. Maurini Annals of Pure und Applied Loyic 89

where the formulas

(1997)

G,, , GP,2, and GP,s, have depth m, the quantifiers

Q=‘, the ones in GP,2 have type @+I,

71

53.-73

in G,,t have type

and those of GP,3 have type Qx,_, . Observe that,

in the formula

GP,2, the free variables _?,X,+, occur only as bounds for the quantifiers, and this means that the formula GP,2 is relativized (for every subscript ,u). Claim 3.2.2.

There exists

the tuples (6, {u +M}),

p such thut the jbrmuh (l?,{v

t-M}),

and j&e

Proof. The formulas @i that we construct any x in N and i= 1,2,3)

The formula

is true for one

at each step satisfy the equivalence

@s is thus satisfied by (c, {U +M}),

satisfied by (0, {u+M}),

GP,2(,?,&+,)

of

jar the other.

and not by (0, {U +M}).

there exists a subscript p such that G,,(g),

G&i?,

(for

Since it is {u+M})

and GP,3({u + M}) hold. The truth of GP,3({u + M}) implies clearly, for every X, the truth of GP,3({x + M}) (because the quantifiers in G,j are Q+,, , and there exists a trivial bijection between [u, cx$ and [x,oo[, for any x). Therefore, G,,(b) and GP,3(o, {V +M}) hold. The truth of formula Glc,2(c, (2: +M}) would imply the truth of @3( 6, {U + M}), and it would contradict our hypothesis. Hence, the (relativized) formula G,, 2 separates the tuples (fi,(~+M}) and (6,{v+M}). q This claim finishes the proof of Claim 3.2.1.

0

Proposition 3.3. For every ordinul a less than CD”,there exists an ordinal cl < x such that the structure (!&(cz), C, <) is a generalized power of’(‘p,(w), C, <) by (‘Q(X), c, <<>. Proof. For every ordinal c(, we denote by B(a) the set that contains 0 and all the limit ordinals iL< X. There exists a unique ordinal E such that E and B(a) are isomorphic (since any subset of an ordinal i is isomorphic to an initial segment of ;I). One can prove by an easy recurrence that the isomorphism is the application cp defined by q(y) = o.y. The ordinal

cl is strictly below IXfor every E < ow (but when x is w’“, the

equality a = ji holds). Step 1: There exists u bijection Qi between the sets !&-(cx) and vf(co)(‘). For any finite subset X of ix, and for any ordinal I -CC, let us denote by comp(X,I) the finite subset of N that contains all the positive integers x such that the ordinal cp(A) +x belongs to X. It is clear that for every 1,
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F. Maurin I Annals of Pure and Applied Logic 89 (1997)

a bijection

between

X of rpf(a),

!&(a)

and ‘$$(cB)“. We shall identify

53-73

in the sequel a finite subset

and its image @x.

Step 2: The relation c is describable. For any finite subsets X and Y of M, the set X is included

in Y if and only if, for any ordinal A -CC?,the set comp(X, A) is included

comp(Y,A).

We take for Fi(x,y)

the formula

and for G the formula (VX)(XcT+X

-(xcy)

= T) (G(T)

(clearly

in

not satisfied by (0,0)),

holds if and only if T is the empty

set). Finally, XcY holds if and only if the structure (5+.$(Z), c, <) satisfies G(T,), when Ti is the set of all the ordinals A -C~7that verify Fi (comp(X, A), comp(Y, A)). Step 3: The relation < is describable. For every finite subset X of a, we denote by Sx the greatest ordinal such that comp(X,S’) is not empty. Let us consider two finite subsets X and Y of a. Then X << Y holds if and only if ,SX is strictly below Sr, or if they are equal and the relation comp(X,Sx) < comp(Y,Sx) holds. Let us introduce the following {c, <<}-formulas (they are clearly not satisfied by s’).

Fl (x, y): (x <
consider

A 7VIDE(y),

F3(x,y): lVIDE(x).

the formula

G( c, Tz, T3):

T3 <
3.4. For every ordinal LY-CO“-‘,the structure (!&-(cl), C, <<) is bounded

Proof. We iterate Theorem

2.1, Section 2, with the help of Propositions

We have to make sure that the triangle

inequality

3.1 and 3.3.

is satisfied by all the norms that

we consider. The norm N, on cP,(o) (defined by N,(X) = supX) satisfies trivially the triangle inequality. It is easy to see that, if the norm N,- (on !@(a)) satisfies the triangle inequality, and if a is such that (‘$,(cr), c, <) is a generalized power of (!$!tr(w), c,<) by (‘pf(@, c,<), then the induced verifies the latter inequality as well. Let P’(U) be the property: “for every ordinal A
norm iV, (defined

on (pf(c~))

the structure (‘P,(A), C, <) is inequality”. Transfinite induca norm on 5&(O), we consider

our main result:

Theorem 3.5. For any ordinal c1
13

F. Maurin I Annals of Pure and Applied Logic 89 f 1997) 53-73

Proof. As Feferman

and Vaught observed in [lo], the structure (w’, +) is a generalized

power of (w, +) by (!&(ly), C, <), for any ordinal CI (there exists a bijection between (u’ and o(‘), and the ordinal addition, viewed as a ternary relation, is describable). It is well known remarks,

that the structure

and appeal to Theorem

(0, +) is bounded

[7]. We then use the previous 3.4. 0

2.1, Section 2 and Proposition

As a final remark, we shall mention

here an application

of the “generalized

power”

tool to natural numbers. We show in [12] that the structure of positive integers with multiplication and restriction of the order < to the set of prime numbers is a generalized power of (N, +) by (!&-(N), C, <). It follows from the present results that the theory of the first structure is decidable and admits a quantifier elimination. This result is to be opposed to the undecidability of TH( N, x, < ), established by J. Robinson in 1949 [14].

Acknowledgements The author thanks Patrick Dehornoy

for his help during the preparation

of this work.

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