On the structure of nonarchimedean

exp onential elds I

Salma Kuhlmann

  

Given an ordered eld K we compute the natural and skeleton of the

ordered multiplicative K in terms of those of the ordered additive

group K We use this computation to provide necessary and sucient

K for the L equivalence of conditions on the value group v K and residue eld



the ab ovementioned groups We then apply the results to exp onential elds and

describ e v K in that case FinallyifK is countableorapower series eld we

derive necessary and sucient conditions on v K and K for K to b e exp onential

In the countable case we get a structure theorem for v K

Intro duction

Given an ordered eld K let v b e its natural valuation with value group G

v K and residue eld K Let v b e the natural valuation of G Classical results

G

on the mo del theory of valued elds have successfully shown how to analyze ele

mentary prop erties of a valued eld through those of its value group and residue

eld One of the aims of this pap er is to do something analoguous for exp onential

elds

What axioms should we require for an exp onential Ahinttoanswer this

question is to think ab out the b est known exp onential eld namely the ordered eld

of the reals endowed with the exp onential function expx In view of the recent

progress in the mo del theory of this structure cf W DMM RE MW

and the pro of of the mo del completeness of its elementary theory ThR expx

it is desirable to give concrete descriptions of the mo dels So understanding the

of exp onential elds is imp ortant since it will provide metho ds

for the construction of examples as we shall see later

In this article we shall rst consider the simplest case an exp onential eld

here is a pair K f where K is an ordered eld and f is an order preserving group



This pap er represents some results of the authors do ctoral thesis

This pap er was written while the author was supp orted by a research grant from the

university of Heidelb erg

isomorphism from the additive group of the eld onto the multiplicative group of

p ositive elements Of course f will also have to induce some extra structure on G

K ifwewanttodoany reasonable analysis of K f So for instance we and on

would like f to induce a canonical map f on K given by f af a This is

equivalent to demanding that

v f

whichinturnisequivalent to the assumption that f maps the valuation ideal I

v

onto the group of units I cf Lemma So wewillalways require to

v

b e satised in an exp onential eld although is not an elementary sentence in

the language of ordered elds it may b e replaced by an elementary but stronger

one cf section Exp onentials satisfying further axioms such as the growth

and the Taylor axioms will constitute the sub ject of the subsequent pap er KK

K f foragiven K f It is more compli Wehaveseenabovehow to obtain

cated to do the same with the value group and it requires rst an understanding of

of p ositive elements where the structure of the multiplicative group K

K is an arbitrary ordered eld not necessarily endowed with an exp onential This

anlysis is done in section and it turns out to b e very fruitful indeed in its

applications to exp onential elds The necessary preliminaries are given in chap

ter The crucial notion app earing there is that of the skeleton S G of an ordered

Ab elian group G The imp ortant result of section is Theorem which com

putes the natural valuation and the skeleton of K in terms of those of

K and G

In section we then apply the results of section to exp onential elds

where K andK are isomorphic through an exp onential f and

thus have the same skeleton since this is an invariant for ordered Ab elian groups

Using the computation of Theorem we are able to show that the existence

of an exp onential f puts a very restrictive condition on the value group of K

whichwe will saytobeanexp onential group cf our denition preceding to

Theorem In particular f induces in a canonical way an isomorphism of

chains

G

f

where v G nfg denotes the value set of G nfg We are now able

G

to formulate the following more concrete problems to which this pap er and the

subsequent pap er KK try to provide an answer

Study the structure and elementary prop erties of K f through those of

K f andG

f

Assume that G is an exp onential group and K admits an exp onential f when

is it p ossible to lift it to an exp onential f of K

Given an archimedean exp onential eld E e satisfying go o d prop erties eg

e is continuous dierentiable and equal to its own derivative give a construction

of a nonarchimedean exp onential eld K f satisfying the same go o d prop erties

and such that K E and f e

In the case where K is a nonarchimedean countable eld ro ot closed for p ositive

elements we get a surprisingly simple result Indeed not only we are able to give

an answer to the ab ove problems but moreover we get quite a strong structure

theorem for those elds cf Theorem In fact given an exp onential f on K

such a eld K admits an exp onential f lifting f if and only if G is isomorphic to

K taken the lexicographic sum of copies of the additive ordered group

over the rationals Theorem will also provide an answer to the third problem

as is shown in KK

The interest of this theorem is that it reduces the construction of nonarchimedean

countable exp onential elds to that of archimedean ones Indeed given a countable

exp onentially closed subeld E of R just take a countable valued eld having E

as its residue eld and E as its value group denotes lexicographic sums

Q

A main ingredient of the pro of of Theorem is a result of R Brown on

countable valued vector spaces anytwocountable valued vector spaces with iso

morphic skeletons are isomorphic Now let us remark that in order to compute the

skeleton of K we prove that for every ordered eld wehave

S I S I

v v

in other words the valuation ideal seen as an additiv e ordered group and the

multiplicative group of units always have the same skeleton cf Corollary

Hence by Browns Theorem weknowthatifK is any countable ordered eld ro ot

closed for p ositiveelements which is equivalent to the assertion that K

b e divisible then it admits a right exp onential ie an isomorphism

f I I

v v

of ordered groups Here the word right is suggested by the lexicographic decom

p ositions K A q A q I and K B q B q I given in

v v

Lemma and Theorem NowifinadditionG K thenwe can nd an

Q

isomorphism h between the left parts A and B of these groups The middle

parts A and B are taken care of by f The lexicographic pro duct of h f and

f will then b e the required exp onential

This principle of decomp osing those groups lexicographically and then showing

that the resp ective parts are isomorphic in order to put an exp onential together

will also b e used several times in KK to obtain a strengthening of Theorem

The left parts are always taken care of by our conditions on the value group G and

the middle parts by conditions on the residue eld K A nal remark concerning

question ab ove an extensive mo del theory for the value groups of exp onential

elds is now given in KF and KF Also the relation b etween the exp onential

eld and its residue eld in the ominimal case is bynow studied in DL At this

p oint I would like to thank FV Kuhlmann for many useful discussions I would

also like to thank Arne Ledet and NL Alling for a hint concerning these notes

and all other participants of our seminar for their interest and patience

Valued and ordered mo dules

Generalities ab out valued mo dules

All mo dules considered in this section are left Rmo dules for a xed ring R The

denitions and results of this section also cover the case of valued Ab elian groups

since they may b e considered as Zmo dules If fM i I g is a family of mo dules

i

L

then M will denote the direct sum

i

iI

In the sequel let M be a module and a chain ie a totally ordered set with

last element A surjectivemap

v M

is a valuation on M and M visa valued mo dule if for all x y M and r R

the following holds

i v x if and only if x

ii v rx v xif r

iii v x y minfv xvy g

Axiom ii says that the scalar multiplication by nonzero elemen ts preserves the

value So we might sp eak of a valued mo dule with value preserving scalar multipli

cation in contrast to mo dules equipp ed with a map v which only satises axioms

i and iii There are also imp ortant applications of the latter more general no

tion of a valued mo dule Both notions will b e considered in subsequent pap ers see

KF In this pap er we will only need valued mo dules satisfying axiom ii so we

will suppress the sp ecication with value preserving scalar multiplication Note

that axiom ii together with axiom i implies that M is torsion free

The following is a consequence of the ab ove axioms

v x v y v x y minfv xvy g

We abbreviate n fg by and call it the rank of M The restriction of v to a

submo dule of M is a valuation on that submo dule

Let M v M v betwovalued mo dules and

h M M

an isomorphism of Rmo dules Wewillsaythath preserves the valuation or that

h is an isomorphism of valued mo dules if there exists an isomorphism of chains

such that for all x M

v x v hx

We will say that M v and M v are isomorphic as valued mo dules if

such an isomorphism h exists Similarly h is an emb edding of valued mo dules if

h is an isomorphism of valued mo dules of M onto a submo dule of M We will

omit of valued mo dules and as valued mo dules if the context is clear Two

valuations v and v on M are called equivalent if the identity map on M is an

isomorphism b etween the valued mo dules M v M v andM v M v

Remark The isomorphism h M M preserves the valuation if and only

if the map

h

rk

given by

h v x v hx

rk

is well dened and an isomorphism of chains

By an ordered system of mo dules we mean a pair

fB g

where fB g is a family of mo dules indexed bysuch that B if

and only if

Let S fB g b e an ordered system of mo dules for i

i i i i

WewillsaythatS and S are isomorphic if there exists an isomorphism

of chains and for every an isomorphism

B B

of mo dules Then wewillcall f gan isomorphism and write

f gS S

Let and put

M fx M v x g

M fx M v x g

Then M M are submo dules for satisfying M M M We put

B M M M B M

Wewillsay that B M isthecomp onent corresp onding to Theskeleton of

M v denoted by S M is the ordered system fB M g We will

write B instead of B M if the context is clear and in what follows B

i

instead of B M for i

i

For every the co ecient map corresp onding to is the canonical homo

morphism

M M

M B dened by x x M

M

and we write instead of if the context is clear

The following lemma shows that an isomorphism of twovalued mo dules induces

an isomorphism of their skeletons

Lemma Suppose that h M M is an isomorphism of valuedmodules

Then for al l the map

h B B h

rk

denedby

M M



x h hx

rk

is an isomorphism of modules Hence

S M S M h fh g

rk

Q

Let fB g an ordered system of torsion free mo dules and B

Q

the pro duct mo dule If s B let

supp ort s f s g

L

The direct sum B is the submo dule of all elements with nite supp ort

We dene

M

v B

min

by v s minsupp ort sbyconvention min This is a valuation and

min

the valued mo dule thus obtained is the Hahn sum denoted by B The

Q

Hahn pro duct denoted by H B is the submo dule of B consisting

of all elements with well ordered supp ort equipp ed with the valuation v We

min

have

a

A

S B fB g S H B

Supp ose that M M and WewillsaythatM v isan extension

M v ifv xv x for all x M In this of M v and write M v

case for every there exists a natural identication of B with a subspace

of B In this context if n we set by convention M M M

M M M and B If and for all B B

we will say that the extension is immediate

Remark The extension M v M v isimmediate if and only if for all

x M there exists y M suchthat v x y v x For example

a

B H B

is immediate

We will saythat M is maximally valued if it do es not admit any prop er imme

diate extension

i I g Let fx i I gM and M M a submo dule Wewillsay that fx

i i

is linearly valuation indep endentover M if for all z M and r R such that

i

r for almost all i I

i

X

v r x z min fv x vz g

i i i

fiI r g

i

iI

If this is the case then in particular fx i I g is linearly indep endentover M

i

By convention is valuation indep endentover M Wewillsaythat fx i I g is

i

linearly valuation indep endent if it is valuation indep endentover M fgNote

that this denition is given for the case of valued mo dules with value preserving

scalar multiplication For the general case it is to o strong and should b e suitably

adapted In the theory of valued elds there is also the notion of algebraically

valuation indep endent but here we will only deal with the ab ove dened notion

so we will omit the sp ecication linearly

Using Zorns Lemma it may b e shown that in every valued mo dule there exist

maximal valuation indep endent subsets

If fx i I gM then

i

M

hfx i I gi

i

R

will denote the Rsubmo dule of M generated by the elements x byconvention

i

M

hi If the context is clear we will omit M orR

R

In the sequel we will assume that R K is a eld so that M V is a valued

vector space with S V fB g For the pro ofs of the facts that

we will state now Lemma till Corollary we refer the reader to the article

GRA of Gravett

Lemma If fx i I gV is maximal valuation independent then

i

hfx i I gi V

i

K

is an immediate extension and

a

hfx i I gi B

i

K

as valuedvector spaces

A basis B of V is called a valuation basis if it is a valuation indep endent set

In this case B is maximal valuation indep endent cf Corollary b elow and

since B admits a valuation basis Lemma gives us a characterization

of those vector spaces which admit a valuation basis

Corollary A valuedvector space admits a valuation basis if and only if it is

isomorphic to the Hahn sum taken over its skeleton

The notion of valuation indep endence is treated in more detail in the following

section

The next theorem is central in the theory of valued vector spaces

Theorem Suppose that

i V and V are valuedvector spaces and V is an immediate extension of V for

i i

i i

i

ii h is an isomorphism of valuedvector spaces of V onto V

iii V is maximal ly valued

Then there exists an embedding h of valuedvector spaces of V into V such that

h prolongates h Moreover h is an isomorphism of valuedvector spaces of V

onto V if and only if V is maximal ly valued

One consequence is the following theorem which illuminates the role played by

the Hahn sums and pro ducts

Theorem Let fx i I gV be maximal valuation independent and h the

i

isomorphism of hfx i I gi onto B Then there exists an embedding h

i

K

of V into H B prolongating h

As a corollarywe obtain a characterization of maximallyvalued vector spaces

Since H B is maximally valued cf GRA the ab ove theorem yields

Corollary A valuedvector space is maximal ly valued if and only if it is iso

morphic to the Hahn product over its skeleton

Also the next lemma is a consequence of the foregoing theorems

Lemma The fol lowing assertions areequivalent

V is maximal ly valued and admits a valuation basis

B H B

B H B

every wel l ordered subset of is nite

is the inverse of an ordinal

Valuation indep endence

Prop osition Let BV Then B is valuation independent over V if and

only if the fol lowing holds for al l n N and dierent b b B with v b

n

V V

v b the coecients b b in B V arelinearly

n n

independent over B V

Pro of Supp ose there are b b B with v b v b such

n n

V V

that b b inB V are not linearly indep endentover B V

n

P

V

Then there exist nonzero k k K such that k b B V If we

n i i

P

V V

cho ose z V satisfying z k b thenwe obtain

i i

v k b k b z min fv b vz g

n n i

fiI k g

i

h shows that B is not valuation indep endentover V whic

P

Let k b b e a nite sum of elements b B with k K and let z V

i i i i

iI

Let min fv b gIfv z then

i

fiI k g

i

X

vz g v k b z v z min fv b

i i i

fiI k g

i

i

Assume now that v z Without loss of generality let n b e precisely

V V

the indices for which k and v b If b b are linearly

i i n

indep endentover B V that is

n

X X

V V V

k b z k b z

i i i i

i

iI

then

X

v fv b g min fv b vz g min k b z

i i i i

fiI k g fiI k g

i i

iI

This prop osition shows

Corollary Let BV Then B is maximal valuation independent over V if

and only if for every v V

V

B f b b B and v b g

forms a basis of B V over B V

A useful consequence of this corollary is the following well known fact It may

b e used to proveby induction that every valued vector space of countable dimension

admits a valuation basis cf BR

Lemma Let V v be a valued K vector space If W is a nite dimensional

subvector spaceof V having valuation basis B andifa V then B can be extended

to a valuation basis of W Ka

Pro of The value set of W is just fv b j b Bg and thus nite Hence there

exists some a W suchthat v a a v W or if this is not p ossible such

that v a a v W is maximal Since the case a W is trivial wemay assume

a W which yields that v a a Ifv a a v W then Bfa a g is the

required valuation basis of W Ka Now assume that v a a v W By

W

the preceding corollary B forms a basis of B W If a a would lie in

B W then there would b e a linear combination a of the elements in B with value

W

such that a a a But this would mean that v a a a

W

a contradiction to the maximalityof This shows that a a cannot b e

an elementof B W In view of the ab ove prop osition we nd that a a is

valuation indep endentover W and again it follows that Bfa a g is the required

valuation basis of W Ka

This pro of shows that the lemma works as well if the condition W is nite di

mensional is replaced byevery subset of v W admits a maximal element or

equivalentlyv W istheinverse of an ordinal cf K

Valuation indep endent sets may serve to obtain isomorphisms b etween valued

vector spaces on the basis of the following lemma

Lemma Let V v and V v be valued K vector spaces containing V v

asacommon valued subspace and such that v V v V Let BV and B V

be valuation independent over V Suppose that there exists a bijection

h BB

B

such that

b B v h b v b

B

Then h extends linearly to an isomorphism over V

B

V V

h hB V i V i hB

of valuedvector spaces which also preserves the valuation moreprecisely which

satises h id

rk

V V

Pro of WeputW hB V i and W hB V i Then B and B are

valuation bases of W resp W over V and h W W is given as follows

P

for every z V and k b a nite linear combination of elements b in B

i i i

iI

X X

h k b z k h b z

i i i B i

iI iI

B and B are valuation bases of W resp W over V and since v h b v b Since

B

for every b B byhyp othesis wehave

X

v k h b z min fv h b vz g

i B i B i

fiI k g

i

iI

z g min fv b v

i

fiI k g

i

X

v k b z

i i

iI

This shows that h preserves the valuation whichinturnyieldsthat h is welldened

and bijective

Generalities ab out ordered mo dules

In this section let R b e a xed commutative ordered ring with satisfying

R

r R nfgs R rs

Note that this condition is satised byevery ordered eld and byevery archimedean

ordered ring Let M be an Rmo dule and a dened on the set M

We will say that M isanordered R mo dule if its underlying additive

R

group is an ordered Ab elian group that is if the order satises

x y M x and y x y

xy if and only if y x

and if the scalar multiplication satises

r R x and r rx

R

Simple consequences are the following rules

if xy then x zy z for all z M

if xy then rx ry for all r R with r

R

x if and only if x

Every ordered R mo dule is torsion free Every ordered Ab elian group is an

R

ordered Zmo dule

Let M M b e ordered R mo dules and

R

h M M

a homomorphism of mo dules We will say that h preserves the order or that h is a

x y M x y implies hx hy homomorphism of ordered mo dulesifforall

A submo dule N M is said to b e convex in M if it satises if x x N

and x M suchthat x xx then x N The set of all convex submo dules of

M ordered by inclusion is a chain containing and M and is closed under unions

and intersections If N is convex in M the quotient mo dule MN equipp ed with

the order induced by

x y x N y N

is an ordered R mo dule

R

Note that if h M M is a surjective homomorphism of ordered mo dules

with kernel N then N is convex in M and h induces an isomorphism of ordered

mo dules from M N onto M

For x M we dene

C M fC C is convex and x C g

x

D M fD D is convex and x D g

x

and by convention

D M fg

A convex submo dule of M is called principal if it is of the form C M for some

x

x M Note that if N is convex in M and x N then

C N C M and D N D M

x x x x

The ordered mo dule

B M C M D M

x x x

will b e called the comp onentof x in M We will write B C and D if the context

x x x

is clear Note that B M fg

We put jxj maxfx xgLety M we will saythat x is Requivalentto y

R

and write x y if there exists r R suchthat

r jxjjy j and r jy jjxj

R

Wewillsaythat x is Rinnitely smaller than y and write x y if r jxj jy j for

R

all r RWe remark the following prop erties is an equivalence relation and

R

is compatible with this equivalence relation

R R

R

y and x z z y x

R R

R

x y and y z x z

The equivalence class of x will b e denoted byx and the set of equivalence classes

R

byWe dene on an order in the following way

R

y x if and only if x y

R R

By the ab ove mentioned prop erties is a chain with last element whichwe

R

will denote by From condition on the ring R it follows that for every r R

R

wehave rx xNow the pro of of the following prop osition is straightforward

Prop osition The map

R

v M

x x

R

is a valuation on M

R

We will call v the Rnatural valuation on M Whenever we consider an

ordered mo dule as a valued mo dule it will b e understo o d that the val

uation is the Rnatural valuation unless otherwise stated Note that a

given map v from M onto some chain is a valuation and equivalenttothe Rnatural

valuation of M if and only if

R

v x vy x y for all x y M

Let v be anyvaluation on an ordered mo dule M v is called compatible with

the order on M if for all x y M with xand y v x vy implies yx

R

In particular v is compatible with the order on M

R R

Wewillsay that M is Rarchimedean if M v ishomogeneousthatisv M

R

contains only one element apart from Ifx M with v x thenC M

x

and D M in particular M is Rarchimedean if and only if M do es not

x

contain anynontrivial convex submo dule Hence B B C D is an R

x x x

archimedean ordered R mo dule and the homomorphism preserves

R

the order To indicate the existence of the induced order on the comp onents we

will sp eak of the ordered skeleton of M and write S M and we will call B

the Rarchimedean comp onent corresp onding to Similarlywewillsay that

S M and S M are isomorphic or that S M andS M are isomorphic

as ordered skeletons if there exists an isomorphism

of chains and for every an isomorphism

B M B M

of ordered mo dules And so on for all other notions dened in section

If M and N are ordered mo dules then M q N denotes the sum M N equipp ed

with the We will not distinguish b et ween external and internal

sums but we will o ccasionally indicate internal sums by writing M M q N

instead of M M q N

Lemma a Let M beanorderedmodule C aconvex submodule of M and

C acomplement to C in M ie C is a submodule of M such that M C C

Then M C q C as orderedmodules

b Let M N a surjective homomorphism of orderedmodules and assume that

ker hasacomplement in M Then

M N q ker

Pro of a It is easy to verify that

M C C C q C

b c b c

is an isomorphism of ordered mo dules

b Let C b e a submo dule of M such that

M C ker

C exists byhyp othesis hence bya

M C q ker

ker b eing convex and

n

j C C N

is an isomorphism of ordered mo dules

For the computation of skeletons there is an analogue to part b of the pre

ceding lemma which has the advantage that it will not require the existence of

a complementtoker in M We rst need some notations If are two

chains then will denote their sum that is the set fg fg

ordered lexicographicallyNow let S fB g b e ordered systems

i i i i

of mo dules for i Then S q S called the sum of S and S will denote the

ordered system

fA g

where

B if fg

A

B if fg

Lemma Let M N be a surjective homomorphism of orderedmodules

Then

S M S N q S ker

Moreprecisely if v and v are the natural valuations of M and N respectively and

the restriction of v to ker is again denotedbyv then the map

w M v N v ker

given by

a if a ker v

w a

v a if a ker

is wel ldened surjective and equivalent to the natural valuation v on M Further

B M B N if a ker

a

a

B M B ker if a ker

a a

Pro of Since the epimorphism preserves that is a b a b its

kernel is a convex submo dule of M It also yields that sends convex submo dules

of M to convex submo dules of N and that the preimage of a convex submo dule

of N is a convex submo dule of M IfC C ker are convex submo dules of

M suchthat C C then C C ker C Wehavethus proved that

C C is a bijective corresp ondence of the convex submo dules of M containing

ker to the convex submo dules of N The set of convex submo dules of M is

linearly ordered by inclusion so the convex submo dules which do not contain ker

are themselves contained in ker Consequently the convex submo dules of M not

containing ker are precisely the convex submo dules of ker

Everything that wehave said holds as well for the principal convex submo dules

Since the corresp ondence C v x is an order reversing bijection from the set

x

of principal convex submo dules of an ordered mo dule onto its value set our ab ove

considerations prove that the valuation w dened by is equivalent to the natural

valuation v on M

By what wehave said ab out the convex submo dules of ker it is clear that

B M B ker whenev er v ker If a ker then the minimal convex

submo dule C of M containing a prop erly contains ker The maximal convex

a

submo dule D of M not containing a will then also contain ker and thus we

a

may compute

B M va C D C ker D ker C D

a a a a a a

By the corresp ondence that wehave describ ed at the b eginning of this pro of it

follows that C is the least convex submo dule of N containing a and that

a

D is the largest convex submo dule of N not containing a This shows that

a

C D B N v a which completes our pro of

a a

Since every convex submo dule C of M app ears as a kernel of some homomor

phism namely the canonical epimorphism M MC the ab ovelemmamay

also b e read as a lemma ab out convex submo dules In the lexicographic sum A q B

of ordered mo dules the submo dule B is convex hence the lemma also shows that

the skeleton of a lexicographic sum A q B is isomorphic to the lexicographic sum

S A q S B of their skeletons This can even b e shown for innite lexicographic

sums

The following two lemmas will b e used in the present and the subsequent pap er

KK Their pro ofs rely signicantly on the foregoing lemma

The lexicographic sum A q B of orderedmodules is maximal ly valued Lemma

if and only if both A and B are

Pro of If B is not maximallyvalued then there is an immediate extension

B of B and by virtue of the foregoing lemma also A q B will b e an immediate

extension of A q B A similar argumentworks if A is not maximally valued

Let M beanontrivial extension of A q B Then the convex hull B of B in

M is an extension of B and the canonical epimorphism M MB induces an

emb edding of A into MB If b oth A and B are maximallyvalued then at least

is not immediate and it follows from one of the extensions B B and A MB

the foregoing lemma that A q B M cannot b e immediate

Lemma The lexicographic sum A q B of orderedmodules admits a valuation

basis if and only if both A and B do

Pro of Set M A q B and denote by the canonical epimorphism onto A with

kernel B

M A

a b a for all a A b B

We shall use the notation of Lemma with A in the place of N The following

fact will b e used in the pro of since B is convex in M wehaveforevery x M

x B if and only if v x v b for some b B

Let f x i I g be a valuation basis for M and set

i

B fx i I g B

i

P

If b B and b r x where all but a nite number of the r s is zero then

i i i

iI

v b min v x

i

fiI r g

i

Consequently for all i I with r v x v b so byx B This shows

i i i

that B is a valuation basis of B

Now let A fx i I g nfg Clearly A is a generating set for AMoreover

i

P

it is valuation indep endent Indeed if r x is a nite linear combination of

i i

elements of Athen

X X

r x v r x v

i i i i

since is a homomorphism On the other hand in this last sum x by

i

P

denition of A so x B Consequently v r x min v x v b for all

i i i i i

P

b B by It follows again by that r x B Nowwe can apply

i i

Lemma to obtain that

X X

v r x w r x min w x minv x

i i i i i i

i i

which proves the assertion

LetA resp B beavaluation basis of A resp of B Then AB is a basis of

P P

M Moreover if r a resp s b is a non trivial nite linear combination

i i j j

iI j J

of elements of A resp of B then by Lemma

X X

A

w s b w r a

j j i i

j J iI

so

X X X X

A

w r a s b w r a v r a minv a min w a

i i j j i i i i i i

i i

iI j J iI iI

where the last two equalities hold by assumption and Lemma It follows that

AB is valuation indep endent

Let us now consider isomorphisms of ordered mo dules If h is such an isomor

phism then it preserves the valuation and induces an isomorphism of the ordered

skeletons But it is the converse to this statement which is imp ortant for us

Prop osition Suppose that M and M areorderedmodules and that

h M M

is an isomorphism of valuedmodules If

h fh g S M S M

rk

is an isomorphism of ordered skeletons then h preserves the order

R M

Pro of Let x and put v x Then x inB M hence

M M



h x That is h hx inB M h whence

rk rk

hx

As a corollarywe obtain the following lemma due to R A H Gravett GRA

Lemme Gravett states it for the case R Q but the lemma is true for arbitrary

ordered Rmo dules

Lemma Suppose that

i M and M areordered Rmodules for i

i

i

ii M M are immediate extensions for i

i

i

n

iii h M M is an isomorphism of valued Rmodules and h j M M

M is an isomorphism of ordered Rmodules

Then h is an isomorphism of ordered Rmodules

n

are immediate Pro of Since h j M M M preserves the order and M M

i

i

extensions wehave that

S g S M fh h M

rk

is an isomorphism of ordered skeletons hence h M M preserves the order

by virtue of the foregoing prop osition

Wewant to exploit this lemma to obtain the analogues to Corollary The

orem Corollary and Theorem in the case of ordered v ector spaces To

this end we need the following denitions Let fB g b e a system

L

of Rarchimedean ordered R mo dules On B we dene the lexico

R

graphical order

l

L

s for all s s B s s if s s for min supp ort s

l

The ordered R mo dule obtained in this way is called the lexicographical sum

R

Q

B consisting of Similarly the lexicographical pro duct is the submo dule of

all elements with wellordered supp ort equipp ed with the order Then for all

l

s s

R

s if and only if min supp ort s min supp ort s s

Hence as valued R mo dule the lexicographical sum resp the lexicographi

R

cal pro duct is isomorphic to the Hahn sum resp Hahn pro duct and we will also

B resp H B Wehave denote it by

a

A

S B fB g S H B

as ordered skeletons

From nowonwe will consider an ordered eld K and an ordered K

K

vector space V In this situation wehave the analogue to Corollary

Prop osition Anorderedvector space admits a valuation basis if and only if

it is isomorphic as orderedvector space to the lexicographical sum over its ordered

skeleton

Pro of Only isnontrivial Let B fx i I g be a valuation basis of V

i

and let

a

h V B V

b e the map given by x s where the tuple s is dened by

i i i

V K

x if v x

i i

s

i

otherwise

and

X X

h k x k s for k K

i i i i i

Here h is the isomorphism of valued vector spaces whose existence is stated in

Corollary Calculating h and fh g one easily veries that

rk

a

h fh g S V S B V

rk

is an isomorphism of ordered skeletons Hence by virtue of Prop osition h

preserves the order

As a corollary to Theorem and Lemma weobtain

Theorem Suppose that

i V and V areorderedvector spaces and V is an immediate extension of V for

i i

i i

i

ii h V V is an isomorphism of orderedvector spaces

iii V is maximal ly valued

Then there exists an embedding h V V of orderedvector spaces such that h

extends h Moreover h is an isomorphism of orderedvector spaces if and only if

is maximal ly valued V

As a corollarywe obtain the analogue to Theorem

Theorem Let fx i I gV a maximal valuation independent subset

i

and h hfx i I gi B V an isomorphism of orderedvector spaces

i

K

Then there exists an embedding h V H B V of orderedvector spaces

extending h

As well we obtain the analogue to Corollary

Corollary Anorderedvector space is maximal ly valuedifandonlyifitis

isomorphic as orderedvector space to the lexicographical product taken over its

ordered skeleton

L equivalence of ordered vector spaces



Let L b e an arbitrary rst order language A B Lstructures and A a common

substructure cf CK for these notions We refer the reader to BAR or POI

for the denitions of the innitary language L the innitary equivalence of A

and B over A whichwe will indicate byA B over A the lo cal or

partial isomorphism the karpian family as well as for the main results concerning

these notions

Following the notation in BAR we will write

I A B over A

p

if I is a nonemptykarpian family of isomorphisms over A of substructures of A

onto substructures of B

Let us state here two results Theorem in BAR and Theorem in POI

of whichwe will make constant use in this pap er For the notion of saturation

app earing in the second theorem see POI

Theorem If A and B arec ountable or countably generated structures then

AB if and only if A B In fact if I A B and f I then f can be

p p

extended to an isomorphism f AB

Theorem If A and B are elementarily equivalent and saturated then

A B

For the pro of of the next theorem see K or K The second assertion of part

a whichwas proved byRBrown cf BR is a sp ecial case of the rst assertion

by virtue of the preceding theorem

Theorem Let K beanordered eld and V V two ordered K vector spaces

a

V V as ordered K vector spac es

if and only if

S V S V as ordered skeletons

If moreover dim V for i then

K i

V V as ordered K vector spaces

if and only if

S V S V as ordered skeletons

b Suppose that S V S V as ordered skeletons Then as

chains and for al l there is some such that B B as

ordered K vector spaces

Applications to exp onential elds

In this chapter we will deal with ordered Ab elian groups which are app earing in

connection with exp onential elds They maybeviewed as ordered Zmo dules

and we disp ose of the notions Zequivalent Znatural v aluation Zarchime

dean and ordered skeleton intro duced in section Since there is no danger of

confusion we will abbreviate the terminology by omitting the Z For example

an ordered Ab elian group is archimedean if it do es not contain prop er nontrivial

convex and wehave

Theorem Every archimedean group is isomorphic as an orderedgroup to a

of R

Cf Fuchs FU for a pro of Hence the skeleton of an ordered Ab elian group is

an ordered system of subgroups of R

The natural valuation of an ordered eld

Let K b e a eld G an ordered Ab elian group and an element greater than every

elementof G A surjectivemap

v K G fg

is a valuation on K if and only if for all a b K

i v a if and only if a

ii v ab v a v b

iii v a b minfv avbg

As immediate consequences wehave

v a v a

v a v a for a

v a v b v a b minfv avbg

We write G v K and call G the value group of K Thevaluation ring is the

ring

fa a K and v a g R

v

and the valuation ideal is its maximal ideal

I fa a K and v a g

v

The eld R I denoted by K is the residue eld The group of units of the

v v

valuation ring is the subgroup of the multiplicative group of R dened by

v

U fa a K and v a g

v

Let nowK b e an ordered eld The set of p ositive elements

of K will b e denoted by K Then K andK are ordered

Ab elian groups and K is divisible Consider the ordered set G of equiv

alence classes a for a K a oftheequivalence relation Zequivalent

archimedean equivalent in the usual terminology dened on the divisible or

dered Ab elian group K the order on G given by

a b if and only if b a

On Gwe dene the addition a b ab equipp ed with this addition and

order G b ecomes an ordered Ab elian group with neutral elemen t and the

natural valuation on the divisible ordered Ab elian group K

v K G fg

a a

isavaluation of the eld K whichwe call the natural valuation on the ordered

eld K It is compatible with the order which means that

if a and b then v a vb b a

In this case R and I are convex in K as ordered additive groups R and I

v v v v

are just C the smallest convex subgroup containing and D the largest convex

subgroup not containing resp ectively Hence K equipp ed with the canonical

order is an archimedean ordered eld and K is just the divisible ordered Ab elian

group C D equipp ed with the multiplication

a D b D ab D

The co ecient map corresp onding to v

K v R

v

is a homomorphism of ordered rings and it is just the residue map of the valued

eld K v To abbreviate the notation we will denote v aby a and omit

corresp onding to v Wehave

a b R a b a b and ab a b

v

In this pap er we will always consider the natural valuation v on an

ordered eld and we will use without explicit mentioning Hence R

v

and I are always convex subgroups of K As a eld K has multiplication

v

and an easy argument then shows that all archimedean comp onents are isomorphic

Lemma The archimedean components of the divisible orderedAbelian

group K are al l isomorphic to the divisible orderedAbelian group K

Pro of Let a K a The map

C K

a

xa x

is a surjective homomorphism of ordered groups with kernel D

a

Remark An isomorphism of ordered elds preserves the natural valuation and

induces isomorphisms of the corresp onding residue eld and value group Hence

K and G are invariants for an ordered eld

The divisibilityofK enables us to present it as a lexicographic sum

of three summands which will play an indep endent role in the course of this pap er

By the the convexityof R and I Lemmas and we obtain

v v

Lemma There exist a group complement A to R in K and a group

v

complement A to I in R such that

v v

K A q A q I

v

Both A and A are unique up to order preserving isomorphism and A is isomor

phic to the archimedean group K Furthermore the value set of A is

G fg the one of I is G fg and the nonzerocomponents of A and I

v v

are al l isomorphic to K

The skeleton of K  

For the multiplicative group K of p ositive elements wewillnow derive

a similar decomp osition as wehave done for the additive group cf Theorem

b elow But the multiplicative group is in general not divisible So as a hyp othesis

in Theorem we will require the divisibilitywhich actually is equivalent to the

prop ertythatK is ro ot closed for p ositive elements for every a K a and

n

for every n N there is some b K suchthat b a Note that every real closed

eld has this prop erty

Let us consider the following subgroups of U the group of p ositive units

v

U fa a andv a g

v

and the group of units

I fa v a g

v

Remark

i By for every a

a R a U

v v

ii For all a b K v a b va implies a signb

iii For every a I wehave v a v a

v

iv If a U n I then a n I U

v v v v

The following two lemmas will give information on the summands of a lexico

graphic decomp osition of K cf Theorem in the case where this

group is divisible But the lemmas are true without this hyp othesis and will yield

a decomp osition of the skeleton of K inany case In view of we

have

Lemma The map

K G

a v av a

is a surjective homomorphism of orderedgroups with kernel U It fol lows that

v

U is convex in K and that

v

K U G

v

as orderedgroups

Next we consider U

v

Lemma The map

U K

v

a a

is a surjective homomorphism of orderedgroups with kernel I It fol lows that

v

I is convex in U and

v

v

U I K

v

v

is a jump ie thereisnoconvex subgroup C such that Consequently I U

v

v

 

I C U

v

 

v

Pro of Follows immediately from the prop erties of the residue map together with

K is archimedean The last assertion follows from the fact that

By virtue of Lemmas and w enow obtain

Theorem If K is divisible then there exist a group complement B

to U in K and a group complement B to I in U such

v

v v

that

K B q B q I

v

Every group complement to U in K is isomorphic to G and every

v

group complement to I in U is isomorphic to K

v

v

In the sequel we will compute the natural valuation w and the ordered skeleton

of K in dep endence up on the natural valuation v and the ordered

G

skeleton fB gofG on the one hand and up on the natural valuation

v and the ordered skeleton of K on the other hand Although the rst

part of the preceding theorem requires the divisibilityofthemultiplicative group

of p ositive elements of K wemay use Lemmas and to compute its sk eleton

even if divisibility do es not hold by means of Lemma

Corollary For every ordered eld K

K q S I S K S G q S

v

The value set in S K consists of just one element apart from since

K is archimedean Consequently the only comp onentinthisskeleton

K apart from fg is itself isomorphic to

Our goal is now to give more detailed information on the valuation and the

comp onents of K For this we need some notations As we are working

with three dierent ordered Ab elian groups for the smallest convex subgroup

containing x and the biggest convex subgroup not containing x we will write

C and D for x K

x x

C and D for x K

x x

C and D for x G

x x

In K for we will write and resp ectively

Nowwe are able to extract some more information from Lemmas and

again by means of Lemma Note that for every a K wehave v v a

G

v v a

G

Lemma

a Suppose that a b K and v a Then b a if and only if v v b

G

v v a and

G

C D C D

a a

v a v a

b Suppose that a b U n I Then a b and

v

v

C D K

a a

since the latter group is archimedean

Now it remains to consider I We will relate its natural valuation

v

to the natural valuation v of the additive group of K and we will showthatits

skeleton is isomorphic to the skeleton of I

v

Lemma Suppose that a and b Ifv a v b then a b

Pro of Weset a and b By hyp othesis there exists n such

that n and n We write

n

X

n

n i

n

i

i

But

n

X

n

i

i

i

hence

n

n

n

Similarly one shows that

Lemma Suppose that a and bLet v b Ifv a vb

then a b

Pro of We set a and b Wewanttoshow for every n

n

Wehave that for every i

i

v iv v v

and thus

n

i

v v

i

whence

n

X

n

i

v v

i

i

In particular by

n

X

n

i

i

i

On the other hand

n

X

n

i n

i

i

hence

n

The following corollary will serve us to describ e the natural valuation of I

v

Corollary If a b and v b then

a b if and only if v a vb

Consequently the map

a v a

is equivalent to the natural valuation on I

v

Pro of The rst assertion is an immediate consequence of Lemmas and

The second assertion is a consequence of the rst by virtue of Remark iii

Lemma For every a I the assignment c D c D establishes

v a a

an isomorphism

C D C D

a a a a

of orderedgroups

Pro of Assume that a I We dene

v

C C D

a a a a

c c D

a

Let us remark that by denition of C and Corollary c C if and only if

a a

v c v a Hence is well dened and surjective Similarly c D if

a a

and only if v c va Hence Ker D

a a

It remains to show that is a homomorphism Given c d C wehaveto

a a

prove that cd c d ie that

a a a

cd c d D

a

This is equivalent to

v c d va

But this is true since

v c d v c v d v a va

Finally preserves the order if c then c

a

From Corollary and Lemma we can deduce the following results

Theorem For every ordered eld K

S I S I

v v

Pro of For v the natural valuation on I Corollary shows that w dened by

v

w av a is equivalent to the natural valuation on I The comp onentof

v

I corresp onding to w a a I isC D this follows from Lemma

v v a a

in view of the convexityofI But by Prop osition C D is isomorphic

v a a

to C D which is the comp onentof I corresp onding to v a w a This

a a v

proves that the skeletons are isomorphic

Corollary Let K beanordered eld root closed for positive elements Then

I I as orderedgroups If moreover K is countable then K admits

v v

an isomorphism f I I of orderedgroups that is K admits a right

v v

exponential

Pro of Since K is ro ot closed for p ositiveelements I is a Q vector space On

v

the other hand by Corollary it has the same skeleton as I So our assertions

v

follow from part a of Theorem

y sequence and pseudo limit app earing in the For the notions of pseudo Cauch

next lemma cf GRA Here will denote ordinals and a limit ordinal

Lemma Let K be any ordered eld Asequence fa g is pseudo

Cauchy in I if and only if fa g is pseudo Cauchy in

v

I Moreover a I is a pseudo limit of fa g if and only if

v v

a I is a pseudo limit of fa g

v

Pro of Let Then by Corollary

a a a a

w w if and only if v v

a a a a

The latter is equivalentto

a a a a

v v

a a

Since v a v a this in turn is equivalentto

v a a va a

that is

v a a va a

This proves the rst assertion The second assertion is proved similarly

Avalued vector space is maximally valued if and only if every pseudo Cauchy

sequence has a pseudo limit cf GRA For the case of valued elds this was

originally shown by Kaplansky KAP For the notion of a p ower series eld ap

p earing in the next corollary and a pro of of the fact that such a eld is maximally

valued see RIB Here let E denote anyarchimedean ordered eld and G any

ordered Ab elian group

G admits a right exponential Corollary Every power series eld E

Pro of Since E G is maximallyvalued its additive group is maximallyvalued

and consequentlysoisI by Lemma Hence it follows from the

v

foregoing lemma that I is maximallyvalued as well As a p ower series

v

eld E G is henselian with resp ect to v By general valuation theory this implies

that its group I of units is divisible By Theorem and Corollary it

v

now follows that the ordered groups I and I must b e isomorphic

v v

Note that this corollary can also b e deduced in a dierentway using a result of

B H Neumann cf N as observed by Alling in ALL section pp

The main theorem of this section now follows immediately from Corollary

Lemma and Lemma

Theorem Dene a valuation

w K G fg

on the orderedgroup K as fol lows w and for al l a

v v a if aU

G

v

w a if a U n I

v

v

v a if a I

v

Then w is equivalent to the natural valuation on K ie

a b K a b w a wb

For A the component of K corresponding to w K we have

B if

A

K if G

K if G

Exp onential elds

Let K b e an ordered eld WewillsaythatK is an exp onential eld

if there exists

f K K

suchthat

f is an isomorphism of ordered groups

v f

A map f with these prop erties will b e called an exp onential on K Note that for

the denition of exp onential eld condition is in fact sup eruous b ecause if

K admits a map e satisfying then it admits also a map f satisfying

and let a K such that ea andput f x eax This pro cedure

also shows a p ossible wayhow to cop e with the fact that axiom is not an

elementary axiom if we do not add a symbol for the valuation to our language

Indeed we are free to replace axiom by the stronger but elementary axiom

f In the place of we could cho ose any other

As well we could taketwo rational numbers r r and use the axiom

r f r Finallybya scheme of axioms of this sort f maybexedto

anyrealnumb er up to addition of an innitesimal an element of value v

Lemma Let

f K K

be an isomorphism of orderedgroups Then the fol lowing areequivalent

v f ie f is an exponential

the map

f K K

a f a

denes an exponential on K

f R U and f I I

v v v

v

Pro of We rst show

Assume holds By Lemma the archimedean comp onent corresp onding to

inK is K R I and by part b of Lemma that

v v

I corresp onding to f in K isU

v

v

So by Lemma the map

f K U I

v

v

a I f a I

v v

is an isomorphism of ordered groups Nownotethat

I R I K U

v v v

v

in fact

b U b I b bI b I

v v v

v

since v b for all b U

v

So indeed wehave that

f af a

Assume now that holds It is immediate to see from the prop erties of f that

must hold

If holds then for every a R wehave v f a and thus v f a

v

So implies

a R n I v f a

v v

Since R n I we see that holds as required

v v

Remark Supp ose that f is an exp onential on K Then the multiplicative

group of K is divisible since it is isomorphic to the additive group of K Hence

there is a decomp osition according to Theorem From Lemma we

knowthatf R U Hence A f B is a group compleme ntto R

v v

v

in K Again from Lemma weknow that f I I Hence

v v

A f B is a group complementto I in R With these groups A and A we

v v

have a decomp osition Denoting the restriction of f to A by f the restriction

L

to A by f and the restriction to I by f wehave obtained isomorphisms

M v R

f A B

L

f A B

M

f I I

v R v

In view of the display of the lexicographic sums this motivates the following de

nition for every ordered eld K with decomp ositions and

An isomorphism f from a group complement A to R in K onto

L v

a group complement B to U in K will b e called a left exp onential

v

in view of the uniqueness of the group complements it automatically induces an

isomorphism b etween A and G Converselyevery isomorphism b etween A and G

induces a left exp onential

An isomorphism f from a group complement A to I in R onto a group

M v v

complement B to I in U will b e called a middle exp onential in view of the

v

v

uniqueness of the group complements it automatically induces an isomorphism

between K and K ie an exp onential on K Conversely

K induces a middle exp onential e A B every exp onential e on

An isomorphism f from I onto I will b e called a right exp onential

R v v

Given a left exp onential f a middle exp onential f and a right exp onential

L M

f then f q f q f is an exp onential on K where

R L M R

a A a A I f q f q f a a f a f a f

v L M R L M R

K and e the corresp onding middle ex In particular if e is an exp onential on

f ethatis f p onential then f f q e q f is an exp onential satisfying

L R

lifts e

Every left exp onential and thus also every exp onential f induces an isomor

phism

G

f

n



more precisely is dened by given by f j

f f rk

G

v a w f a v v f a

f G

This may b e indicated by writing w f v Indeed by Lemma the

f

group A has G fg as its value set and the isomorphism f A G thus

induces the ab ov e isomorphism of the value sets More precisely f even induces an

isomorphism of the skeletons of A and G so in view of Lemma all comp onents

of G will b e isomorphic to K

Every right exp onential and thus also every exp onential f induces an auto

morphism

G G

f

n



given by f j Itmay b e indicated by writing w f v which

f rk f

G

is dened by v w f v f This is b ecause G fg is

f

the value set of b oth I and I Again f induces also an isomorphism of the

v v

skeletons but indep endently of the existence of f wehave already sho wn that the

skeletons of I and I are isomorphic cf Corollary

v v

As a corollary to Lemma and the preceding remark we obtain the following

result which is due to Alling cf ALL Th and Cor

Corollary If K admits an exponential f then

G as chains

K for every B

the map

f K K

a f a

is an exponential of K

Note that and of this corollary already hold under the assumption that K

admits a left exp onential

This last corollary motivates the following denition Let G b e an ordered

Ab elian group with skeleton fB g and denote by v the natural

G

valuation on G so v G nfg Further let A b e an archimedean ordered

G

Ab elian group Then G will b e called an exp onential group in A if

it admits a group exp onential ie an isomorphism G of chains

B A for all

With this denition the statement of the preceding corollary reads as follows

Theorem If K f is an exponential eld then G is a divisible expo

f

nential group in K and K f is an exponential eld

Again note that the rst assertion of this theorem already holds under the as

sumption that f is a left exp onential on K

Another consequence of the ab ove remark is the fact that there are only very

sp ecial extensions of an exp onential eld which are immediate with resp ect to the

natural valuations Let L v jK v b e an extension of valued not necessarily

exp onential elds Then K v issaidtobedense in L v ifforevery a L and

v L there is some b K suchthat v a b We rst need the following

characterization of densityLetA b e a group complement for the valuation ring

K

of K v Using Zorns Lemma it may b e extended to a group complement A for

L

the valuation ring of L v

Lemma K v is dense in L v if and only if A A If this is the case

K L

K L that is the extension is immediate then v K v L and

Pro of If K v is dense in L v then for every a L with v a there

is some b K such that v a b which implies that A is already a group

K

complement for the valuation ring of L v

For the converse assume that the latter holds Let a L and v L Then

v K since otherwise an element d L with value v djj could not

b e represented as the sum of an elementof K and an elementofthevaluation

ring of LSowemaycho ose some c K with v c Byhyp othesis for ac

there exists some b A such that v ac b Putting b b cwe nd

K

v a b v c This proves the converse

If is chosen va then v a b va implies v a v b v K

a b K Hence if K v is dense in L v then wehave and if v a also

v K v L and K L

Theorem Suppose that K f L f is a nontrivial extension of nonar

chimedean exponential elds Let v denote the natural valuation on both ordered

elds and denote the ranks of v L resp v K Then either K v is

L K

dense in L v or n is innite

L K

Pro of Let A A as ab ove If A A then the foregoing lemma shows

K L K L

that K v is dense in L v Now assume that A A is a prop er extension

K L

Comp osing the exp onential f with the canonical epimorphism given in Lemma

we then nd that also v K v L is a prop er extension and thus v L n v K

is innite Through the isomorphism it follows that n is innite

f L K

Wehave seen in Corollary that every p ower series eld which is ro ot closed

for p ositive elements admits a rightexponential As a further consequence of the

ab ove remark let us state here a necessary and sucient condition for sucha eld

to admit a left exp onential

Prop osition Let E G beapower series eld root closed for positive el

ements If E G admits a left exponential then G is maximal ly valued more

precisely

G H E



G

Consequently G does not admit a valuation basis Conversely if G is exponential

in E and maximal ly valued then E G admits a left exponential

Pro of Since E G is a p ower series eld it is maximally valued That is the

additive group of E G with its natural valuation is maximally valued Assuming

the decomp osition it then follows by Lemma that A is maximallyvalued

If E G admits a left exp onential then G is isomorphic to A and thus also

maximally valued Hence G will b e a Hahn pro duct over its skeleton according

to Corollary Since G is an exp onential group in E by Theorem

holds On the other hand G implies that contains innite well

ordered subsets and by Lemma it follows that G do es not admit a valuation

basis

If G is exp onen tial in E and maximally valued then G is of the form

hence G A and this induces a left exp onential cf Remark

Corollary Let E G beapower series eld root closed for positive ele

ments Let e beanexponential on E Then E G admits an exponential lifting e

if and only if G is exponential in E and maximal ly valued

Pro of follows from Prop osition

By Prop osition E G admits a left exp onential On the other hand

by Corollary it admits a right exp onential The assertion nowfollows from

Remark

Toprove our next theorem we need Lemma b elow which is based on the

following result cf Fuchs FU for a pro of

Lemma Let H J be an isomorphism of orderedsubgroups of R Then

there exists an element r R such that a r a for every a H

Lemma Let H J bearchimedean orderedgroups If H J as ordered

groups then H J as orderedgroups

Pro of Wemay supp ose that H and J are subgroups of R cf Theorem Let

H J b e a lo cal isomorphism of a subgroup H H onto a subgroup J of

J Then there exists r R such that a r a for every a H Hence

wehave that r H J Let us showthatr H J Let a H and H b e the

subgroup of H generated by H fagand

H J

a lo cal isomorphism such that Then there exists s R such that

a s a for every a H Itfollows that r s and hence r a J

By a symmetrical argument considering

J H

b r b

it maybeshown that r J H ieJ r H Itfollows that J r H and

hence

H J

a r a

is the required isomorphism

Remark

There are two essential ingredients for the pro of of our next theorem

i We will use Theorem In fact K andK b eing divis

ible ordered Ab elian groups we consider them as ordered Q vector spaces It is

imp ortanttoverify here the following general fact Let G G b e divisible ordered

Ab elian groups then G G as ordered groups if and only if G G

as ordered Q vector spaces Let us briey sketch a pro of of the nontrivial direc

tion Let F G G as ordered groups it suces to nd F G G such

p p

that for every f F domf and im f are divisible ordered Ab elian groups ev ery

isomorphism of divisible ordered Ab elian groups b eing an isomorphism of ordered

Q vector spaces The construction of F from F is done without diculty as fol

lows If f F dom f H imf H andifH resp H denote the divisible

closures of H resp H inG resp G then f extends in a unique waytoan

isomorphism

f H H

and we take

F ff f F g

ii Every dense linear ordering without endp oints is saturated Indeed the

theory of dense linear orderings without endp oints is complete and admits elimi

nation of quantiers cf RZ Consequently realizing a typ e over a nite set

of parameters reduces to solving nitely many inequalities of the form x a and

xb But this is always p ossible in such an ordering

It follows by Theorem that anysuch ordering is L equivalentto Q

Converselyifa chain is L equivalentto Q then it is necessarily a dense linear

ordering without endp oints

We will further use the notations of Theorem

Theorem Let K be a nonarchimedean ordered eld such that K

is divisible Then K K as orderedgroups if and only if

G K as orderedgroups

Q

K admits an exponential

Pro of By part a of Theorem S K S K as

ordered skeletons By Theorem and part b of Theorem we obtain that

i G G as chains

ii for every B K as ordered groups

iii K K as ordered groups

K since all archimedean comp onents of K are isomorphic to

But K b eing divisible G is also divisible and hence as a chain is a

dense linear ordering without endp oints let us remark that G since K is

nonarchimedean By i it follows that G is a dense linear ordering without

endp oints so the same holds for By part ii of the preceding remark it follows

that

Q

Let Q Since by ii and Lemma all nontrivial archimedean comp o

p

nents of G are isomorphic to K it is easy to construct from a family

a

K S G S

p

Q

So assertion follows now from part a of Theorem and assertion follows

from iii and Lemma

Let us assume the decomp ositions and see Lemma and Theo

rem Let e b e an exp onential on K and e the induced middle exp onential

e A B

By Corollary weknowthat

I I

v v

So our assertion will followassoonas

A B

is proved In fact if F A B and F I I then

L p R v p v

F F qfe gq F K K

L R p

where F qfe gqF ff q e q f j f F f F g cf denition

L R L R L L R R

But v A G is a dense linear ordering without endp oints so

G Q

On the other hand all comp onents of A are isomorphic to K so byan

argument already used in the rst part of this pro of it follows that

a

S A S K

Q

It follows by part a of Theorem that

a

A K

Q

But B G so together with the hyp othesis of the theorem we obtain that

a

B K A

Q

As a corollarywe obtain the following theorem whichgives us the converse

to Corollary if we replace by

Corollary Let K be a nonarchimedean ordered eld such that K

is divisible Then K K as orderedgroups if and only if

G as chains

B K for every

K admits an exponential

Pro of By arguments already used in the pro of of the previous theorem one

can see that conditions and together are equivalent to condition of the

theorem

Note that by part ii of Remark assertion is equivalent to the prop ertyof

to b e a dense linear ordering without endp oints Note also that by virtue of

Lemma the corollary is true also for archimedean elds

For the countable case the ab ove theorem yields

Theorem Countable Case Characterization Theorem

Let K beacountable nonarchimedean ordered eld such that K is divis

ible Given an exponential e on K itcan be liftedtoanexponential f on K ie

f e if and only if G K

Q

K are countable Pro of follows from Theorem and the fact that G and

by virtue of Theorem

Given an exp onential e on K lete and F b e as in the pro of of Theorem

Soby virtue of Theorem there exists Then every element f of F extends e

an isomorphism f K K extending e It follows that

f e

Note that the condition on G in the ab ove theorem is actually equivalent to the

K in fact wehave the following assertion G is an exp onential group in

Prop osition Let G and A becountable divisible orderedAbelian groups

and let A bearchimedean Then G is an exponential group in A if and only if it is

of the form G A

Q

Pro of Let G be a countable exp onential group in AThenG so

is a countable dense linear ordering without endp oints By a classical theorem

of Cantor it follows that Q as chains On the other hand all comp onents of

G are isomorphic to Asoby part a of Theorem wehave G A

Q

Let G AThen Q andG is a countable dense linear ordering

Q

without endp oints Again byCantors Theorem G Since all comp onents

of G are A it follows that G is exp onential in A

From the pro of of Theorem we obtain the following corollary whichcontains

the symmetrical analogue to Corollary

Corollary Let K beanordered eld root closed for positive elements

a If G is an exponential group in K then G K

Q

b Assume the decompositions and Then A B as orderedgroups if

K If moreover K is countable then K admits an and only if G

Q

isomorphism f A B of orderedgroups that is K admits a left exponential

K if and only if G

Q

Remark In all the preceding theorems and corollaries wemay replace the

function exp onential by K linear exp onential Indeed the app earing value

groups are not only Q but also K vector spaces and Theorem holds for arbi

trary ordered vector spaces This is of interest b ecause also the usual exp onential

on R is in fact Rlinear

To nish this pap er let us saya word ab out the hyp othesis of divisibility for

K In fact Theorem used in the pro of of Theorem is not

true if one omits the hyp othesis of divisibility The following example due to

FV Kuhlmann shows that there exist two regular countable ordered Ab elian

groups having isomorphic skeletons which are not isomorphic as ordered groups

An ordered Ab elian group A is regular if and only if AB is divisible for every

nontrivial convex subgroup B of A

Example In Q Q take a subgroup H such that the pro jection to the rst

Z but whichis comp onentisQ and the pro jection to the second comp onentis

not a direct sum of Q and Z the existence of such a group is well known On

Q Q take the lexicographic order and on H the restriction of this order We

have

HC H Q

which shows the regularityof H The archimedean comp onents are Q and Z

Z implies that but H Q

H Q q Z

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