Some Model Theory for Almost Real Closed Fields - Semantic Scholar

Some Model Theory for Almost Real Closed Fields Author(s): Françoise Delon and Rafel Farré Reviewed work(s): Source: The Journal of Symbolic Logic, Vol. 61, No. 4 (Dec., 1996), pp. 1121-1152 Published by: Association for Symbolic Logic Stable URL: http://www.jstor.org/stable/2275808 . Accessed: 06/12/2012 08:46 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp

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THE JOURNALOF SYMBOLICLoGic

Volume 61, Number 4, Dec. 1996

SOME MODEL THEORY FOR ALMOST REAL CLOSED FIELDS

FRAN(SOISE

DELON

AND

RAFEL

FARRE

Abstract. We study the model theory of fields k carrying a henselian valuation with real closed residue field. We give a criteria for elementary equivalence and elementary inclusion of such fields involving the value group of a not necessarily definable valuation. This allows us to translate theories of such fields to theories of ordered abelian groups, and we study the properties of this translation. We also characterize the first-order definable convex subgroups of a given ordered abelian group and prove that the definable real valuation rings of k are in correspondence with the definable convex subgroups of the value group of a certain real valuation of k.

?1. Introduction. In [14] B. Jacob introduced the notion of Hereditarily SPythagorean fields and studied their model theory. The almost real closed fields include these fields and the results of ?3 may be seen as a generalization of some results of [14] and [15]. By an almost real closed field we understand a field carrying a henselian valuation with real closed residue field. The simplest example of an almost real closed field is a real closed field. The almost real closed fields appear in a natural way when studying algebraic properties of Hereditarily-Pythagoreanfield and dealing with closures in a certain sense: the generalized real closed fields of Becker (see [1] and [2]), chainclosed fields in the sense of Harman (see [13]) and in the sense of Schwartz (see [24]). Another example of almost real closed fields are the Rolle fields of Brown, Craven and Pelling (see [4]), whose model theory was studied by F Delon in [7]. Other characterizations of almost real closed fields are obtained in [23] and [3], where the algebraic properties of such fields are studied. In Section 2 we study the almost real closed fields (ARC) and the chain W(k) of real valuations of such a field k. We make special attention to vo, the first element of the chain with real closed residue field. Also the valuations vs (the first element with S-Euclidean residue field), for S a set of primes, play an important role. Their rings are the Jacob rings defined in [14] and turn out to be first-order definable in the case where S is finite. We prove that the theory of almost real closed fields is elementary and provide an explicit axiom system using the first-order definability of vs for S finite. Received August 31, 1992; revised July 18, 1995. 1991 Mathematics Subject Classification. Primary:

03C60. Secondary: 12D15, 12J10, 06F20, 03B25.. Key words and phrases. Henselian fields, real closed fields, ordered abelian groups, decidability..

Partially supported by Alexander von Humboldt-Stiftung. Partially supported by grants PB91-0279 and PB94-0854 of DGICYT

and PR9014 of UPC. ? 1996, Association for Symbolic Logic 0022-481 2/96/6104-0002/$4.20

1121


1122

FRANCOISE DELON AND RAFEL FARRE

In Section 3 we prove the transfer Theorem 3.5. In fact, one of the implications of 3.5 is a consequence of the Ax-Kochen-Ershov theorem and of the completeness and model-completeness of the theory of real closed fields. The new fact is that the theory of the field k (without valuation) determines the theory of vo(k). As a consequence we prove that there is a bijection between theories of almost real closed fields and certain theories of ordered abelian groups. The theories of ordered abelian groups involved in this bijection are characterized in Proposition 3.10. Moreover, this bijection preserves the completeness and, in certain cases, the decidability. We provide examples showing that this bijection does not preserve in general neither the decidability nor the recursive axiomatizability. If the valuation ring of vo were uniformly definable without parameters, Theorem 3.5 would be an immediate consequence of this fact. This raises the question whether vo has a definable valuation ring. In Section 4 we study the definablereal valuation rings. For this purpose, we start by giving a characterization in 4.1 and 4.2 of the definable convex subgroups of an ordered abelian group. In Theorem 4.4 we prove that there is a bijection between the definable real valuation rings of k and the definable convex subgroups of the value group of the maximum element of W(k). This result allows us to conclude that vo is definable only when vo = vs for some finite S. This criterion does not apply to every valuation vs for S infinite. vs may be definable even in the case it is not equal to any vs, for S' finite. In the case where dimp k < oc (definition in ?2) the definable elements of W(k) are completely characterized. ?2. Almost real closed fields. If (k, v) is a valued field, we are going to denote by AV, MV, UV,v(k), k/v and 2z respectively the valuation ring, the maximal ideal of AV, the group of units of AV, the value group, the residue field of (k, v) and the canonical projection from AVto k/v. If v, v' are two valuations of k and AV C AV, we shall denote it by v > v'. Then v induces in a natural way a valuation on k/v' which we will denote v/v'. If k carries an order < with P its positive cone, v is called convex or compatible with < if AV is convex in k, which is equivalent to 1 + Mv C P. Then < induces an order in k/v by putting 21(a) > 0 if and only if a > 0 for every a E UV.If F is a subfield of k, AF= { xI

< y foracertainy

E F}

is a valuation ring of k with F C AF. We are going to use the notation VF for the valuation associated to AF. Hence 7TF is invectiveon F. If F' is another subfield of k with AF = AF' then TF = 7TF' is invectiveon F' and 7TF(F') is not bounded in (k/VF, v implies v' E V(k).

PROPOSITION2.1.



1123

(iii) If v' is any valuationof k, v E V(k) and v' < v then v' E V(k) if and only if v/v'(k/v') is divisible. (iv) If V(k) is not empty, then it is a final segment of W(k) with minima and maxima. The minimalelement vo of V(k) is the valuationof V(k) with no non-trivial convex divisiblesubgroupin vo(k) and the maximum element v, is the valuationwith archimedeanreal closed residuefield. PROOF.(i) If k is real then W(k) is not empty, and every henselian valuation over k is convex for every order of k [16]. Fix any order of k. The convex subrings of k are totally ordered by inclusion. (ii) and (iii) are proved using systematically the following facts: (1) If v is a real valuation of k, then k is real closed if and only if v is henselian, k/v real closed and v(k) divisible [16]. (2) For two valuations v' > v over k, one has (by [19]) v' is henselian if and only if v and v'/v are henselian, k/v' v(k)

(k/v)/(v'/v), v'(k)/(v'/v)(k/v).

(iv) is a consequence of (i), (ii) and the fact that given an order < of k there exists A a convex valuation v with archimedean ordered residue field. The value group of any valuation of V(k) is the quotient of vl (k) by a convex divisible subgroup, and vo(k) is the quotient of v1(k) by its biggest convex divisible subgroup. Thus, given two valuations v, v' E V(k) and any prime number p, dimp v(k) = dimpv'(k), where dimp v(k) is the dimension of v(k)/pv(k) as IFP-vectorspace. When V(k) # 0, it allows us to speak of dimp k for every prime number p. As we will see, these dimensions by themselves contain much information about the field k. DEFINITION. We are going to call a field

k almost real closed (ARC) if V(k) 7 0.

2.2. Let k be afield. PROPOSITION

(i) If k is almost real closed then so is any real algebraic extension of k. (ii) If v is a real valuation of k then k is almost real closed if and only if k/v is almost real closed and v is henselian. (iii) If k is almost real closed, W(k) = { v v is a real valuationof k }. PROOF. (i) is routine. (ii) If v' E V(k), it is convex for every order of k, hence comparable with every real valuation of k. We now distinguish the cases v > v' and v' > v, and finish the

proof with the same tools as in 2.1 (i) and (ii). (iii) follows from (ii).

A

It is easily seen that k is real closed if and only if it is an almost real closed field and dimp k = 0 for every prime number p. k is a Rolle field if and only if it is an almost real closed field with dimp k = 0 for every odd prime number p ([4]). In a similar way we can characterize chain-closed field and generalized real closed fields.


1124

FRANOISE

DELON AND RAFEL FARRE

Also it can be shown that there are exactly 2dim2 k orders in any almost real closed field k. We are interested in first-order definable valuations of k, and Jacob valuations ([14]) will provide us with important examples of those. In order to understand the meaning of these valuations let us introduce some definitions. Let P be the set of all prime numbers and S C P. Following [14] we call a valuation v over a field k S-Kummer henselian (S-Kh) if Hensel lemma holds for polynomials of type xP - a when p E S and a E Av. p-Kh will mean {p}-Kh. When v is real, then v is S-Kh if and only if for any a E Uv and p E S, a E kV if and only if r(a) E (k/v)P. LEMMA2.3. Let k be anyfield and v, v' two valuations of k such that v' > v and k/v' has characteristiczero. Then v' is S-Kummerhenselian if and only if v and v'/v are.

PROOF.The characteristiczero of the residue fields of v and v' makes the polynomials of type xP -a having only simple roots when a :&0. The only non-trivial step is to prove that v is S-Kh when v' is. If xP - rv(a) has a root 7rv(b), then 7rv(a/bP) = 1 and 7rv(a/bP) = 1. Applying the S-Kh of v' to the polynomial xP - a/bP we find c E k with c = a/bP and 7rv,(c) = 1. We will finish if we prove nrv(c) = 1 because then cb is the desired root of xP - a. This follows from (2Tv(c))l' = 1, lzv'(c) = 1 and the zero characteristic of k/v'. Note that we only need that the characteristic of k/v' does not belong to S. We call a field k S-Euclidean if it is real and k = ?kP for every p E S. If p :&2 this condition is equivalent to k = kV. For S = {p} we are going to say p-Euclidean. The usually Euclidean fields are exactly the 2-Euclidean fields in our terminology. If k is a field we define Vs (k) = { v I v is a S-Kh valuation of k with S-Euclidean residue field }. Vp will denote Vfp}. LEMMA2.4. Let k be afield and v a real valuationof k. Then: (i) v E Vs (k) if and only iffor every x E k and p E S thefollowing equivalence

holds: x E +kP

if and only if v (x) E pv (k).

(ii) If v E Vs (k), m E N* such that all its prime divisorsare in S and x following equivalenceholds: x E +km

ifandonlyif

E

k, the

v(x) E mv(k).

(iii) Thefollowing are equivalent: k is S-Euclidean if and only if v E Vs (k) and v (k) is S-divisible. PROOF.Supposed E Vs(k) andlet a E k suchthatv(a) = pv(b) for acertain b E k. Then v(a/bP) = 0 and thus 21(a/bP) E +(k/v)P. Applying the S-Kh to the polynomial xP -T a/bP we deduce a/bP E ?kP whence a E ?kP. This proves the only if part of (i). For the if part let a E k with v(a) = 0. This implies a E ?kP for p E S and hence 2r(a) E ?(k/v)P and Hensel lemma holds for the polynomial xP - a (k/v has no primitive p-roots of unity if p > 2). This proves



1125

v E Vs (k). (ii) follows by iterating (i). The left to right implication of (iii) follows easily keeping in mind that k/v has no primitive p-roots of unity for p > 2 and A k/v is real. The other side follows from (i). Point (iii) of the preceding proposition suggests us to call a field S-almost Euclidean (S-AE for short) when Vs (k) :&0. Clearly an S-Euclidean field is S-almost Euclidean and every S-almost Euclidean field is real. If k is a real field and v is a 2-Kh valuation then 1 + Mv C k2 and thus v is compatible with any order of k. By the same arguments as for V (k), the set W2(k) of 2-Kh valuations is totally ordered. If 2 E S then Vs (k) C W2(k), whence Vs (k) is also totally ordered. 2.5. Let k be afield and S a set ofprimes. Then: PROPOSITION (i) If v' is a real valuation of k and v E Vs(k), v' > v implies v' E Vs(k) and v'/v E Vs(k/v). (ii) If v' is a valuationof k, v E Vs (k) and v' < v then v' E Vs (k) if and only if v/v'(k/v') is S-divisible. (iii) The infimumof two valuationsof Vs (k) belongs to Vs (k). (iv) If Vs(k) 7&0, Vs(k) has a minimumelement. The minimumelement is the unique valuationof Vs (k) withoutnon-trivialS-divisible convex subgroups. (v) If 2 E S and Vs(k) :&0, Vs(k) is afinal segment of W2(k) with maximum element, i.e., the valuationv1 of Vs (k) with archimedeanresiduefield. PROOF.The proofs of (i) and (ii) use the same arguments as the proof of Proposition 2.1 replacing everywhere henselian by S-Kh and real closed by S-Euclidean and using Lemmas 2.3 and 2.4. CLAIM. If k carries two independent p-Kh valuations of residual characteristic zero, then k = kV. PROOFOF THECLAIM.Let v and v' be two such valuations and a, b E k. From the approximation theorem for independent valuations, there exists c E k such that

v(c-a)

> v(a)

and v'(c-bP)

> v'(bP).

From the p-Kh of v' we deduce c E kV and from the p-Kh of v we conclude A a E k. (iii) If the two valuations v and v' are comparable the conclusion is obvious. If v and v' are not comparable, applying the claim to v/v A v' and v'/v A v' (v A v' stands for the infimum of v and v') we deduce that 2 , S and k/v A v' is S-Euclidean, whence v A v' E Vs (k). This shows again that when 2 E S, Vs (k) is totally ordered. (iv) follows from (iii). (v) rake any v E Vs (k) and any order of the residue field k/v. Take v' the valuation of k/v compatible with this order and with archimedean residue field. Taking v1 the valuation of k composition of v and v' we have vi E Vs (k) and k/vl is archimedean. From the linear order on W2(k) we deduce that it is the only A one.


1126

FRANOISE DELONAND RAFELFARRE

EXAMPLE. Let n > 1 be an integer, k a field with n orders existentially closed [26], and S = IP \ {2}. Then k = k for all p E S, hence

Vs(k) = { v I v a real valuation on k }. If k is a bit saturated, its orders are not archimedean and Vs (k) is not linearly ordered. Indeed two distinct orders of k are independent, which means that they define distinct topologies. Now any non-trivial valuation which is convex for some order defines the same topology as this order. Hence the archimedean valuations associated to two different orders on k are not comparable. We shall denote the first element of Vs (k) and Vp(k) by vs and vp respectively. The valuation rings of these valuations are the Jacob rings (see [14]) defined as follows:

{x

x V ?kP and 1 + x E kV for some p E S }, 62(k, S) = {x E k x E ?kP for all p E S and x691(k,S) C &1(k,5) &i (k, S) =

Ek

},

&(k,S) = 61(k,S) U &2(k,S). REMARK.

When S is finite &(k, S) is definable by a first-order formula of the

language of rings LR = {+,-, I, 0, 1}. PROPOSITION2.6. Let S C P and k be an S-almost Euclideanfield, then M(k, S) is the valuationring of vs.

PROOF.If v E Vs (k) we remark the following facts: (1) x E 61 (k, S) if and only if v(x) > Oand v(x) V pv (k) for some p E S. (2) v(x) > 0 and v(x) E pv(k) for all p E S implies x E 92(k,S). (3) If v(k) has no non-trivial convex S-divisible subgroup then Av = a9(k, S). PROOFOF (3). From (1) and (2) we have that 61(k,S) C Av C 6 (k,S). It remains to prove that if x E 62(k, S) then v (x) > 0. Suppose v (x) < 0; from the fact that v(k) has no non-trivial S-divisible convex subgroups we deduce the existence of y E k such that 0 < v(y) < -v(x) and v(y) is not S-divisible. Thus y E &1(k,S) andv(xy) < 0 whencex V &2(k,S). A As a consequence of Proposition 2.6 we deduce that if S is finite the class of S-AE fields is elementary: take sentences expressing that &(k, S) is an S-Kh valuation ring with S-Euclidean residue field. LEMMA 2.7. Let k be anyfield and S C P. Thenk is S-AE ifand only ifk is S'-AE for every S' E Pf (S), the set offinite subsets of S. PROOF.The left to right direction is obvious. Conversely, suppose Vs, (k) :&0 for every S' E Pf (S), and let As, be the valuation ring of vs,. Then

A

=

n{As,

IS

E Pf (S) }

is a real valuation ring because it is the intersection of an inverse filtrant family of real valuation rings (Aslusl C As/ n Asll). If v is the valuation corresponding to A obviously v > vs, and from Proposition 2.5 we have that v E Vs, (k) for A every S' E Pf (S) whence v E Vs (k).



1127

As a consequence of Lemma 2.7 we deduce that for any S the class of S-AE fields is an elementary class. REMARK.If k is S-AE and S' C S C P then Vs (k) C Vs/ (k) and vsl < vs. From

the fact that the maximum convex S-divisible subgroup of an ordered abelian group is the intersection of all the maximum convex S'-divisible subgroups for S' E Pf (S) it follows that vs = sups, Epf (s) vse and thus &(k,s)

=

n{i(k,SI)

IS

E Pf(S)}

(this is shown in [14] using the definition of a (k, S)). Our interest in the sets Vs (k) is when k is almost real closed. In this case, even when 2 V S, Vs is totally ordered because every real valuation is henselian. Then obviously V(k) = Vp(k) C Vs(k) C W(k)=W2(k)= whence vs < vp = vo < v1 for every set of primes S. 2.8. The class ARC is elementaryand its theory is decidable. PROPOSITION

PROOF.This class is closed by ultraproducts and elementary equivalence: if k is almost real closed and L =- k, from the Keisler-Shelahtheorem k and L have isomorphic ultrapowersk U _ LU; thus LU carries a henselian valuation v with real closed residue field. Because L is relatively algebraically closed into LU, v [L E V(L). Applying the Ax-Kochen-Ershov theorem [27], the theory T of henselian valued fields with real closed residue field is decidable in the language of valued fields. We are using here the decidability of the theory of ordered abelian groups [12] in addition to the decidability of the theory of real closed fields [5]. The decidability A of ARC follows from the decidability of T. An Hereditarily-Pythagoreanfield (do not mistake with the notion of Hereditarily S-Pythagorean in [14]) is a real field k such that every real algebraic extension is Pythagorean (a sum of squares is a square). E. Becker (see [1] for example) has characterized Hereditarily-Pythagoreanfields as those fields k for which every finite extension is of the form k( tlFa1, VdY,. . ., ,) for some a1, ..., an E k and t1 ...,

tn E N-

Property (iii) in the following proposition provides an explicit axiomatization of ARC. 2.9. For anyfield k thefollowing are equivalent: PROPOSITION

(i) k is an almost real closedfield. (ii) (a) every real valuationof k is henselian, and (b) k carries an order < such that for every archimedeansubfield (F, 2.

| H/Hn _ G/Gn

(ii) If H C G, then H - G if and only if

Hn = {O}if and only if Gn = {O} for all n > 2 Hn = Gnn H and H/Hn - G/Gn for all n > 2. PROOF. The implications from left to right follow from the fact that the maximum convex n-divisible subgroup is definable by a formula without parameters, namely x E Gn if and only if

G

F

(Vy) ((x >?

AO < y < x) V (x < O AO < y < -x)

-*

(z)

(y = nz)).

For the converse, if Gn = {O} for some n > 2 the other implications are trivial. If Gn :8 {O} for all n > 2 the same happens for Hn, and using Lemma 3.1 and Theorem C of the Appendix, (i) follows from Sp~H)

SPn (H/Hn) U {a }

Spn (G/G

{

SPn(G),


1130


where a is defined as in Lemma 3.1. We use here the following claim, whose elementary proof we omit. It may be seen as a consequence of [11]. CLAIM. Let A be a coloured chain with minimum, which we are going to denote by 0. Let A* be the coloured chain consisting in adding to A an extra element a (a V 2), placed in the order following the minimum (a > 0, a < b for all b E 2t, b > 0) and satisfying some fixed set of monadic predicates (the same for all such 2). Then, for every formula 0 (xi,.. , xm, Xm+l, . . ., xn) in the language of coloured chains there is a formula p'(x1,... ., xm) in the language of coloured chains such that for any coloured chain A with minimum, and for any a1, . . ., am E A we have

*

ifandonlyif

1=p(aj,...,ama,...,a)

A

=p'(aj,..

am).

In particular, if p is a sentence, p' is a sentence and A* i= p if and only if A i= p'. For the proof of (ii) we use Theorem D of the Appendix. From Lemma 3.1 and the claim above, Spn(H)

SPn(H/Hn) + {a}I

SPn(G/Gn) + {a}Iv Spn(G).

From Hn = Gn n H and H/Hn -< G/Gn we have that nH = nG n H and thus the elementary inclusion Spn(H) ' Spn(G) is defined by SPn (H) AH

SPn(G)

-

(h) - A G(h)

An (h) = {}-

>

Fn'(h)

1

0

if h , Hn

A G = {0}

if h

Hn \ {0}

if h V nH

3~F(h) -0.

It remains to prove that the extension H C G preserves the predicates M (k), CH(h) and E(n,k) and D(p,r,i). If H F M(k)(h) thenh , Hn and CH/Hn(h) thus H l= M(k)(h) if and only if H/Hn Fz M(k)(h). Doing the same for G, from H/Hn -< G/Gn we conclude H l= M(k)(h) if and only if G l= M(k)(h). Also if H l= E (n, k) (h) then h V nH and a similar argument shows that H l= E (n, k) (h) if and only if H/Hn F E (n, k) (h). For D (p, r, i) it suffices to remark that h e prH f -r 14(h). prH if and only if e pr(H/Hp) and if h then (h) pr/HP REMARKS.

(1) Theorem 3.2 is not true if we restrict the conditions to n prime, as the following example shows. Let H = e

G=

Fn and

n o-*

En, nCco*

where co* denotes the inverse order of o and 2 Fn =

if n =0

22

if n is odd

23

ifn >0, n is even

En=

if n =0

Q 23

if n is odd

22

ifn >0, n is even.


1131


Here Z2 and Z3 denote respectively the groups of 2-adic and 3-adic rational integers. It is easily seen that Hp :& {O} and Gp :& {O} for all prime p, H/Hp {O} for all prime p :&2, 3, and G/Gp @ @ @ -

HIH2

GIG2

H/H3

GIG3 c

Z3

Z2

Z2,

3

ED 22

2d 2(

23.

However H 0 G since H2 c H3 and G3 c G2. (2) In fact, as suggested in the example above, the relations Hp C Hq for p, q E P is the extra information we need in order to code Th(G). Theorem 3.2, for elementary equivalence, may be stated as if and only if

H-G

{ Hp

=

=

{O}ifandonlyifGp

{O}

forallp

EP

Hp C Hq if and only if Gp C Gq

for all p, q E P

lH/Hp

for all p E P.

-G/Gp

This follows from Theorem 3.2 since for n > 2 Hn = this happens at the same time in H and G since Hn =

Hp Hp

for some prime pIn and if and only if Hp C Hq

for all prime q, q In. It also holds a similar result for elementary embeddings. NOTATION. For any ordered abelian group G, Go = convex divisible subgroup of G.

np,Gp, the maximum

COROLLARY3.3. Let G and H be two ordered abelian groups. (i) If Ho = {O} and Go = {O}, then H-G

(ii) If Ho H-
2.

H/Hn-=G/Gn

Go = {O} and H C G, then

if and only if

Hn = Gn n H and H/Hn 2.

G implies H/Ho G/Go. (iii) H (iv) H < G implies H n Go = Ho and H/Ho -< G/Go. (v) G G/Go if and only if either Go = {O} or for all p E P there is some q E P such that Gq c GP. (vi) H/Ho - G/Go and G 0 H if and only iffor some p E P, Ho = Hp, = Go Gp, H/HpG/Gp and exactly one of Hp, Gp is equal to {O}. C H G if and only if (vii) If G, then H n Go = Ho, H/Ho j G/Go and H for some p EP, and GP 7 {O}.

Ho = Hp,

Go = Gp, H

G

Hp,

H/Hp

-< G/Gp,

Hp = {O}

PROOF. (i) and (ii) follow from Theorem 3.2 and the following remark. If Go = C GPforallp E Pifandonlyif(G/Gnp)n = {O} {O},thenGn = {O}ifandonlyifGn for all p E P. (iii) follows from Remark 2 to Theorem 3.2 and 3.3 (i) bearing in mind that (G/GO)n = Gn/Go, whence G/Go/(G/Go)n G/Gn (iv) H G implies H n GP = Hp for all p E P whence H n Go = Ho. The rest

of the proof is as in (iii).


1132


(v) (v=) If Go = {O} is clear and the other case follows from Remark 2 to Theorem 3.2 as in (iii), since Gp C Gqif and only if (G/Go)p C (G/Go)q. (v) (=) If Go {O} and for some p C P Gp C Gqfor all q E P then GP# {0} and (G/Go)p= {O} whence G 0 G/Go. (vi) ( =>) If H/Ho - G/Go and H 0 G then either H 0 H/Ho or G 0 G/Go, hence by 3.3 (v) there is some prime p such that {O} :&Ho = Hp or {0} = Go = G. In any case, from H/Ho _ G/Go we get Ho = Hp and Go = Gp, whence H/Hp G/Gp. Hp, Gp may not be equal to {O} nor different from {O} at the same time since by Remark 2 to Theorem 3.2 this would imply G H. The converse is clear since Ho - Hp and Go = Gp. (vii) follows from (vi) bearing in mind that in this case Ho = Hp and Go = Gp.REMARK. If, in the conditions of Corollary 3.3, we want to deal with prime n only, we can set (i) If Ho {O} and Go = {O}, then

G if and only if

H

>HpC Hq if and only if Gp C Gq for all p, q E P |H/Hp -G/Gp for all p E P. (ii) If Ho = {O}, Go = {O} and H C G. then G if and only if

H

{p

C Hq if and only if Gp C Gq Hp Gp n H and H/Hp - G/Gp

for all p, q E P for all p E P.

We are going to denote by LOG = {0, +, -,