REND. SEM. MAT. UNIVERS. POLITECN. TORINO

Vol. 45°, 2 (1987)

Carlo Toffalori

PAIRS OF DISCRETE o-MDMIMAL STRUCTURES

Let T be a (complete) strongly minimal theory. When studying the model theory of the pairs (N,M) with N^.M t= T, one can follow two approaches:

1. to consider the theory T' of all the pairs (N,M) where N ^ M f= T, and to check if T' is complete (for example, a classical theorem of A. Robinson [R] states that, if T is the theory of algebraically closed fields of given characteristic, then Tr is complete; on the other hand, there exist strongly minimal theories T such that T' is not complete, such as the theory T of infinite sets);

2. to notice that T admits a unique non-algebraic 1-type p over 0 (p is stationary, too), and to consider the theory Tp of the pairs (M(a),M) where M \= T, a t= p\M and M(a) denotes the model of T prime over M U {a}. Then it comes out that Tp is anyhow complete, and either Tp = T or the only models of Tp are the pairs (N(a), N) where N 1= T and a f= \=p\N\ the second case happens when the fact "dimMM(^)= 1" can be expressed by a suitable 1st order sentence in the language for pairs of models of T (see [T]). Hence the models (N,M) of Tp are exactly those satisfying

* N — M =fc0 (notice that every element a EN — M realizes the nonforking extension p\M of p in S1(M)),

* the sentence ensuring "dimMN= 1" if this sentence exists, or the fact that this sentence does not exist otherwise.

Classificazione per soggetto AMS (MOS, 1980): 0 3 C 3 5

60

Let T be now a (strongly) o-minimal theory [PSl], and consider again the pairs (N,M) with N^M\=T.

1. In this context the first approach is quite unsatisfactory; in fact, let V be the theory of all pairs (N,M) with N^.M \= T, notice that this time it is very unlikely for T' to be complete; the reason is that there may exist (N,M), (Nr, M') f= T' such that N fills a cut over M which is definable without parameters (such as the one corresponding to + °°), while N' does not fill the corresponding cut over M'\ then (N,Af)f£ (N\ M'). For instance, con-sider the theory T of real closed fields, and let T\ be the theory of the pairs (N,M) where N>M\=T and M, N-M are dense in N, T'2 =Th((IR(0, IR)) where t is an infinite hyperreal and IR(£) denotes the real closure of JR(t). Then both Tj and T2 are completions of T', but T\=fcT2.

2. The second approach seems more practicable. In fact, let M t= T, p be a non algebraic 1-type over M, a\=pf and consider the pair (M(a),M) where M{a) denotes again the model of T prime over M U {a} (see [PSl]). Then the difficulties in 1. — for instance, which cuts over M are filled in M(a)? — can be overcome by looking at p> and the analysis of the whole matter seems plainer. Nevertheless there exist remarkable complications with respect to the strongly minimal case. The first one is that p may not be the only non algebraic 1-type over M, and even it may exist some types q GS1(M) such that q^p but q~Mp (~M denotes here the Rudin-Keisler equivalence relation in S1(M), see [Mr] § 3; notice that [Mr] prefers the symbol AM). Moreover, as T is unstable, "nonforking extensions" make no sense in this context. Hence we need answer some preliminary questions, namely if we can find now, for every N \= T} a type over N playing the same role as p\N does in the strongly minimal case, and, anyhow, if p L is as important as before. The difficulty of these problems recommends a prudent approach; therefore we start from the simplest case, assuming that the order of T is discrete (see [PS2] and [PS3]). First we will deal with the following questions, arising in a natural way from the above remarks:

* for any M 1= T, to describe the non algebraic 1-types over M\

* for any M \= T, p ESi(M) (p non algebraic) and a t= p, to describe p\~M and, for all qEp\~M, q(M(a)).

This is what we are going to examine in § 1; part of the corresponding analysis is implicit in the Pillay and Steinhorn papers, however a general ac-count is necessary as a ground for our aims. § 1 is also devoted to the prob-lem of finding a substitute of p\N when N t= T in this context; this does

61

not cause any difficulty if p is the type of +°° or — °°; in the remaining cases we will see that, if N>M, then p admits only one extension in 5X(7V) if and only if p(N) — 0, and we will denote this extension by p \N.

However the key part of the paper is § 2. Here we will consider a model M of T and a non algebraic pESi(M) such that p\~M ls finite (for instance p\ ~M = {px,... ,pn) where we can assume p=pi M)=T and moreover

* N— M = U (N — M); where, for all ; = 1,... ,w, (N—M\ is non Kj)" HMJ=0;

* 7 = 1 , . . . , » - 1 , Vxe(N-M)j, vy£(N-M)j+1, x

62

"rational type", "irrational type" instead of "cut" and "noncut"; in any case, we will recall the definition of these notions in the next section.

For all a, b EL/ and M t= T, {a,b)M will denote the set {x EM : a

63

happen both a.cl. 0 = 0 (see T = Th (Z ,

64

Fact 1.3 - Let M f= T, p be a non algebraic 1-type over M. Then p is said to be irrational if

* there exist m, m' EM such that m s~i{m).

Then there are 4 kinds of irrational types p over M:

kind 1 — p satisfies (A), (B) ;

kind 2 - / 7 satisfies (A), ~I(B) ;

kind 3 — p satisfies "1(A), (B) •,

kind 4 —p satisfies "1(A), "1(B).

Let M l= T, p G5j(Al), p non algebraic. The problems we want to deal with now are the following:

1. to describe p\~M 5

2. to describe p(M(a)) (where a t= p) and, more generally, q(M(a)) for all qEp\~M.

65

Analysis of the rational case

First we lock at the case p — p^ (p = p^ can be discussed in a similar way). Since, for every irrational qGSi(M), q Y'M p¥°» (see [Mr]), then

PROPOSITION 1.4 - Let M 1= T. Then

P+~~M P-oo

if and only if there exist a O-definable function f and m, tn' such that m, m' E a.cl. 0 or m — — °° and m' = + °°, and / is an order reversing bijection of (w,+o°) onto (—o°,m'). Furthermore, in this case, for all N t= T, P+o* NP-oe-

Proof. ( W J , then x b. Then / L +oe) is order

reversing, and /((m, +°°)) = (~°°, m') where m' E a.cl. 0 if m E a.cl. 0, while m' = + «» if m = —

The second claim is obvious. —

We have to study now p^^iMia)) (and p^JMW) in the case P+oo MP-OT,)-

THEOREM 1.5 - There exists an infinite discretely ordered group G such that, for every M t= T and a t= p ^ , p^oo(M(tf)) is order isomorphic to G.

Proof First notice that, if M f= T and a (= p^L, then, for all b EM(a),

if and only if there exist a O-definable total function / arid m,m' such that m, m' E a.cl. 0 or m = m' = —«», / generates an order preserving bijection of (m,+°°) onto (m',+ oo) and f(a) = &.

In fact («=) is obvious. Conversely let b EM(a), b t= p ^ , , t n e n there exist a O-definable function / such that b = f(a) (see above). We can assume that / is total, by replacing, if necessary, / with the function / ' such that,

66

for all x G U, f'(x)=f(pc) if xEdomf, and f'(x)=x otherwise. In any case / induces an order preserving or reversing bijection of an interval I of the form (m, +°°), with m Ga.cl. 0 U {—°°}, onto a second interval /(/-). Let x EM, x>m, then x m, f f(x) = x, and, for all y>m\

ff'(y)=y-Let now / , g G G, define / ~ g if one of the following conditions holds

(it is easy to see that they are equivalent):

(i) there exists r G a.cl. 0 U {—°o} such that, for every x G U, if x > r, then f(x)=g(x);

(ii) for every Mf=T and a NpJL, /(#) = g(a)\

(iii) there are M f= T and a 1= p^ such that /(a) =g(a).

Then ~ is an equivalence relation in F and, for every / , / ' , g, g' EF,

if f~f and g ~ g' , then /g ~ f'g' .

Similarly the following conditions are equivalent:

(i) there exists r G a.cl. 0 U {—°°} such that, for every x EJJ, if x > r, then f(x)

67

(iii) there are M t= T and a l= p ^ such that f(a)

68

THEOREM 1.6 - For every discretely ordered infinite group G, there is a complete O-minimal theory T of discretely ordered structures without end-point such that, for all M \=T and a f= p^lo, P™oo(M(a)) is order isomorphic to G.

Proof. Let. G be an (additive) discretely ordered infinite group. Denote by u the least positive element in G, and look at N — {nu :n G Z } . Then N is a convex normal subgroup of G, and the factor group G/N is an ordered group, provided we set, for all g, g' E G,

N + g

69

Analysis of the irrational case

Let M \=T, p £ S2 (Af) be a non algebraic irrational type. First we shall deal with the problem of describing p | ~M.

PROPOSITION 1.7 - For every q E5X0W), q ~M p if and only if there are a O-definable total function f and /, / ' O-definable intervals such that / Ep, I' Eg , f\j is an order preserving (or reversing) bijection of / onto V and, for all x E /,

x < v Gp if and only if f(x) < v E q {fix) >v €.q) .

Furthermore, if p is of kind 1, 4, then q is of kind 1, 4, too; if p is of kind 2 or 3, then q is of kind 2 or 3, too.

Proof (M (and, consequently, the restric-tion of p over a model M' of T such that M'

70

PROPOSITION 1.8 - Let Af ) It is trivial.

(

71

Then it is clear that p satisfies (A) -(B)- if and only if p\M' satisfies (A) -(B)-, hence p \M' is of the same kind as p. -

PROPOSITION 1.10 -Le t p, qESx(M) satisfy p~MvEq).

Then f order preserving implies / " order preserving. In particular, if p = q, then both f and / " are order preserving.

Proof. Suppose towards a contradiction that / ' is order preserving, but / " is order reversing. Then, for every x, y EI satisfying x < v M and p(M') = 0, then Gp = Gp^M> .

Proof. First notice that, if M F= T, p is an irrational 1-type over M and a \=p, then, for all b EM(a), b \= p if and only if there are a O-definable total function / , and I, I' O-definable intervals such that /, / ' Ep, f is an order preserving bijection of / onto / ' , f(a) = b and, for all x EI,

v

72

assume b

73

We set in this case f

74

* g(I) is a O-definable interval, g(I)Ep, g is an order preserving bijection of / onto g(I) and, for all x EI, x

75

such that m

76

p ESiiM) (clearly p 3 p 0 ) such that Gp is order isomorphic to S (and

hence, for any a 1= p, p(M(a)) is order isomorphic to S).

1st case: S = G (this is the case when, for example, G ^ Z). Consider

* tEU (then a.cl. t = {s ; '/x (t): j E'E, X G G/AT} is the domain of the model Af0 of T prime over £);

*t'EU,t't=p^.

Let Af =M 0 (O = M 0u (*;/*(*') : / £ ^ , XGG/N}; for all ; € Z and

X E G/N, s'fx(*') 1= / £ £ . Let pESx(M) be defined by

Af 0

77

§ 2. Pairs of structures

Let T be a complete o-minimai theory of discretely ordered structures without endpoints. Recall that we denote by U the universe of T\ in the following, for every ~a, "a' E U,

a =a

will abridge tp(tf"| 0) = tp (a' 10) (in T). We list now some facts which will be useful in the following.

Fact 2.1 - Let M

78

* if p^ fM p^ (recall that in this case p+„ fN p"„ for all N 1= T), then for any x EM(a) —M, x realizes p ^ ;

* if p̂ oo ~M P™~ (equivalently p^ ~ N p"„ for all N h 7), then there are a O-definable total function h and m,m' such that either m,m' G a.cl. 0 or m = — °° and m' == + , m'); moreover, for any t£M(«)-Af, either £1= p j ^ or there is # £M(a) such that a; 1= p^ and £(*;) = £.

Hence let T" be the theory of the pairs (N,M) where N ^ M l = T and

* ^ P+oo'fuP-oo, then, for any xEN—M, x>M\

* if p^oo ~up-oo> then, for every # E N - M , either # > M or x M;

* H'EM' is non empty and algebraically independent, b' EN' is algebraical-ly independent over Af' and >M'-,

* ( a > ) = ( * > ' ) .

Then, for every tEM such that (a,t) is algebraically independent,

79

there exists t'EM* such that (a, b,t) = (a',b,,t')\ moreover (a',t') is algebraically independent.

Let M if and only if / - Ax) > Mf

(recall that H,!? are non empty). Hence we have:

* if x0EM, then x'0 EM' and s(xf0) Ey(M',~a', b')-

* if xxEM, then x\ EM' and 5_ 1 (* i ) €tf>(Af',*',£');

* if ^ 0 , x1^M, then # 0 < M < # 1 , so that ^ Q ^ ^ ' ^ ^ I J anc* ^(W,

It follows that {v EM'} U {^>(v,^', &') : 1/ t= (M, £)} is finitely satisfiable in (N',M')i and hence is realized in {N',M') — say by t' — . Clearly (#", b,t) = (a', b\ t')\ consequently (a', t') is algebraically indepen-dent.

(iii) Let (a,b)t(a',b') be as in (ii). Then for all tEN such that t>M and (b,t) is algebraically independent over M, there exists t'EN' such that (a,b,t) = (a',b',t') (so that, in particular, t'>M' and (b',t') is algebraical-ly independent over M').

Let y(v,~a,b)Evp{t\~a, b), we claim that there is t' Ey(N',~a', b') such that t'>M'. As in (ii) we can assume that $(N,~a,b) is an infinite interval o f jhe form ( ^ 0 , ^ i )

N where x0, x1 E a.cl. (a,b) U {±°°}. Then , x[ -f-^{xx) other-wise. By recalling that t >M, we can distinguish the following cases:

* xx E N , xx >M\ then x\ = / - ^(# j) >Af', hence we can put t' = s- 1 ^ ) ;

* ^ 1 = + ©o> x0EN, X0>M] then ^ i = + o ° , a^EAf', # Q > M ' , and we can choose tf = s (x'Q)\

80

* xx = + oo, but x0 does not satisfy the previous conditions; then, for all t'GN' with t'>M\ we have t' Eip(N',l',~b').

Then the set {v $M'} U {y(v,~a', b'): U $= a' for every a' E.~a') is finitely satisfiable, and hence realized, in (N',M') — say by t' — . Then t' >M' and (#",&, 0 = (a\ b', t'). It is easy to deduce that (b\ t') is algebraically independent over M'.

(ii)' and (iii)' — the '-versions of (ii) and (ii) — can be shown in a similar way.

Let / be the set of all the partial isomorphism f- ^ of (N,M) into (N',M') where (a, b) and (a', b') are as in (ii). Then we have the following facts.

* / =£ 0 (see (i)).

* For all / E / and t E JV, there exists g E / such that / C g and t E dom g.

In fact, let / = /_ r where (a,b), (a', b') are as above. If t E a.cl. (U, £), then put g = / . If £^a.cl. (# ;£) and t EM, then (a, £) is algebraically independent; for (ii), there is t'EM' such that (a,b)t) = (a',b

,,t')\ put 7 = ( a , 0 , 2' = (a',t'), g=f?i', then gGJ, g D / , £Edomg. Finally, if £ ̂ a.cl. (a", b) and £^A1, then there is no loss of generality in assuming t>M (otherwise, consider h~1(t))\ in any case (b, t) is algebraically in-dependent over M, hence, for (iii), there is t'EN' such that (a,b,t) = = (5',b'ft'), t'>M' and (b',.t') is algebraically independent over M'; put ? = (£,*), ~c' = (b',t'), g = / ^ > then gEI, gDf, tedomg (as well as h(t), in case).

* For all fEI and t'GN', there exists gEI such that f(=g and £E Eimg.

Proceed as above, by using (ii)' and (iii)' instead of (ii) and (iii).

Hence / : (N,M) ^ (iV',Af'). -

Finally notice that, if (N,M) is an N0-saturated model of T+08, then, for any finite XCN, (N,M) realizes every 1-type over X (in T); by recal-ling that "algebraic over X" means "X-definable", one can easily deduce that (N,M) realizes every 1-type over a.cl. X. In particular we have:

PROPOSITION 2.4 - Let (N,M) be an N0-saturated model of T+00, a, b EN with (a,b)N infinite;

81

(i) if (a,b)N DM^0, then dim (a,b)N HM is infinite;

(ii) if {a,b)N -M=£0, then dimM(a,b)N is infinite.

Proof, (i) First suppose a, b EM\ then (a,b)N - (a,b)M . Let X be a finite, algebrically independent subset of {a,b)N. Since (a,b)N is infinite, there exists p G5i(a.cl. (X U {a,b})) containing asJ{a) : / e i } U { » EM] in (N,M) ;

otherwise a £M and hence aPn) (where p1 -p). Then, for all / =

= ! , . . . , « , there exist a O-definable total function fj, and Ij, I- O-definable

82

intervals such that Ij E.p, IJE.pj, fj is an order preserving (or reversing) bijection of Ij onto Ij and, for any x E/;-, x v Ep ;) . Moreover, for all / = 1,..., n, fj(a) f= pj. Without loss of generality, we can assume f1 = id, and a

83

Moreover, in this case, n = 2 and one can choose f2 so that f2 gene-rates two decreasing bijections, the one inverse of the other, of pi(M0(a)) onto p2(M0(a)) and of p2(M0(a))

o n t o Pi(M0(a)) respectively.

Proof. First assume (—°°, a) D a.cl. 0 = 0, and fix xx G a.cl. 0 n (A, f2(a)) such that / 2 is an order preserving (or reversing)bijection of (~°° ,^ i ] onto (—°°,/2(^i)] (or 1/2(^1),

+OG)). L e t / i be order preserving. As x1>a, we have f2{xx)>f2{a), hence a^ < / 2 ( x 1 ) ; then there is xE(—/;;(#), it follows #

84

Fix a Odefinable total function f2 satisfying (*), assume ty = f2(t), then one can replace t with the greatest element t' 1, put

7ib =fhUl) »

then J/;) is of the form [x^, y^] where x ,̂, y^ G a.cl. 0, #j, < / j , (0) ,*) n a.cl. 0=^0, (/»(«),+°°) H a.cl. 0 # 0 , (/*(*), fh + i(a)) ^ a-c^ 0 ^ 0 f° r a ^ £ = 1,...,» — 1; /i> is an increasing or de-creasing bijection of pi(MQ(a)) onto ph(M0(a)) for all h = l,...,w.

It is easy to see that all these cases really occur. However, for simplicity's sake, we will continue our analysis only in the third case, which is clearly the

85

most complicated, and, as regards the remaining cases, which can be discussed in a similar way, we will limit ourselves to sketch the basic points concerning them at the end of this section.

Hence, let p satisfy the third case; consider the theory Tp of the pairs (M(a),Af) where M>M0, p(M) = 0 and a\=p\M. Notice that, for all h = l,...1n, ph(M) — 0 and the ~M-class of p\M just equals {px\M,..., pn\M}. Furthermore, for every h, ; = 1,...,«, JhEpj\M if and only if h =;', and f^ is an order preserving or reversing bijection of Jx onto Jf,, and hence of px(M(a)) onto pfy(M(a)). Then consider the theory T' of the pairs (N,M) where N>M\=T and (N,M) satisfies the following axioms (1M3).

(1) 3w0,...,wn 3vx,...,vn ( A I D J G M A A Vi^MA i

86

(3) for all x a.cl. 0 (in T),

xEM0 when * a ;

xEMi (Ki

87

It follows that, for every x G a.cl. (a, b) and h = 1,..., n,

x e (jv-M)h * x S M , x ej% «• /- 5(*) £A*', /5fj(«) eyf • • / f , ; ( * ) € C N ' - A f % .

Similarly, for every x S a.cl. (a, fc),

^ - M , * * e « , x / , ( * ) * / j ( 5(*)€Af' , / J i 5 (* ) > / , ( * ' ) *

while, for every b = 1,...,» — 1,

x S M i • * e M , /»(*)

88

The proof is similar to the previous one.

(ii)' and (iii)' — the '-versions of (ii) and (iii) — can be shown in the same way.

Consider at this point the set I of the partial isomorphisms of (N,M) into (N', M') of the form / - ^ with (a,b), (a', b') as in (ii). Then we have the following facts.

* I =£ 0 (see (i)).

* For all fEI and tEN, there is gEI such that fQg and tEdomg.

In fact, let / = / j £ . ^ t Ez.cl. (a,b), then pick g = / . Otherwise, distinguish three cases.

1st case: t EM. Then t $• a.cl.

89

PROPOSITION 2.9 - Let (N,M) be an N0-saturated model of Tp,a, b EN satisfy (a,b)N infinite. Then

(i) if (a,b)N nM¥=0, then dim (a,b)N DM is infinite;

(ii) if (a,b)N — M =£0, then d i m ^ ^ , ^ ) ^ is infinite.

As we already noticed in the introduction, these results present a sharp analogy with the ones concerning strongly minimal theories.

Finally, we owe some comments on the cases 1 and 2 of the irrational analysis. It comes out that formally the results of the case 3 (Theorem 2.8, Proposition 2.9) still hold. Hence the only point to be clarified is what T' means (in other words, how to axiomatize Tp) in this context. However, after carefully studying (M0(a),M0)f it is not difficult to find the corres-ponding list of axioms.

REFERENCES

[B] J. Barwise, Back and forth through infinitary logic, MAA Studies in Mathematics 8(1973), 5-34.

[CK] C.C. Chang-H.J. Keisler, Model theory, Studies in Logic 73, North Holland (1973).

[Mk] M. Makkai, An introduction to stability theory, with particular emphasis on orthogonality and regular types, Israel J. Math. 49 (1984), 181-238.

[Mr] D. Marker, Omitting types in o-minimal theories, J. Symbolig Logic 51 (1986), 63-74.

[PS1] A. Pillay-C. Steinhorn, Definable sets in ordered structures I, Trans. Amer. Math. Soc. 295 (1986), 565-592.

[PS2] A. Pillay-C. Steinhorn, Discrete o-minimal structures, Ann. Pure Appl. Logic, to appear.

[PS3] A. Pillay-C. Steinhorn, Definable sets in ordered structures III, preprint.

[R] A. Robinson, Solution of a problem of Tarski, Fund. Math. 47 (1959), 179-204.

[ST] C. Steinhorn-C. Toffalori, The Boolean spectrum of an o-minimal theory, pre-print.

[T] C. Toffalori, Pairs of structures, Quaderni Istituto Matematico "U. Dini" (Firenze) 4(1986-87).

90

CARLO TOFFALORI - Istituto Matematico Universita Viale Morgagni 67/A - Firenze (Italia)

Lavoro pervenuto in redazione Vll/V/1987