On Types over p-adically Closed Fields

Ningyuan Yao Zhentao Zhang

Abstract. The aim of this paper is to study types over a p-adically closed field.We classify the 1-types over an arbitrary p-adically closed field,which extends the previous work of Penazzi,Pillay and Yao(2019)on classifying 1-types over the standard model Qp of the field of p-adic numbers.We also study the orthogonality of pseudo-limit types and distance types and yield an analogue of the dichotomy of “cuts” and “noncuts” in the o-minimal context.

1 Introduction

This paper presents several new results on types inp-adically closed fields,the structures(in the language of rings)which are elementarily equivalent to the field Qpofp-adic numbers.We denote the theory of Qpin the language of rings bypCF.

Delon showed in[3]that every type over Qpis definable.In[10],the authors classified the complete 1-types over the standard model Qpas follows:

Theorem 1.[10]The complete1-types overQp are precisely the following:

(i)The realized typestp(a/Qp)for each a ∈Qp;

(ii)For each a ∈Qp and C,a coset ofinGm,the type pa,C saying that x is infinitesimally closed to a(i.e.v(x-a)>n for each n ∈N),and x-a ∈C;

(iii)For each coset C as above the type p∞,C saying that x ∈C and v(x)

whereGm the is multiplicative group of a very saturated elementary extension ofQp,and

In this paper,we extend the above result to an arbitraryp-adically closed field:

Theorem 2.Let K be a model of pCF,ΓK the value group of K,andas above.Then the complete1-types over K are precisely the following:

(i)The realized typestp(a/K)for each a ∈K;

(ii) (distance type around a point)For each cutΛ⊆ΓK(see Definition 2),c ∈K,and coset C of,the type pΛ,c,C saying thatΛ

(iii) (Pseudo-limit type)For each pesudo-Cauchy sequence{ci}i∈I(see Definition1),the typesaying that x is a pseudo-limit of{ci}i∈I,and for any formula φ(x)over K,φ(x)∈iff φ(x)is eventually true(see Definition1)on{ci}i∈I.

Note that whenKis Qp,any pseudo-limit type is realized since Qpis complete as a metric space.As the value group of Qpis Z and there are exactly two cut over Z,namely,∅and Z,we see that case(ii)of Theorem 2 corresponds to case(ii)and(iii)of Theorem 1.

Recall that ano-minimal structure is an ordered structure (M,<,...) in which every definable subsetX ⊆Mis a finite union of intervals and points.IfA ⊆Mandc ∈M,we call tp(c/A)a cut iff there area,b ∈dcl(A),the definable closure ofAinM,such thata

In thepCF environment,it is reasonable to consider the pseudo-limit types and distance types as the analogues of “cut” and “noncut” in theo-minimal context,respectively.

Assumingo-minimality,a result of[8]shows that if tp(c/A)is a type of cut overA,and tp(d/A)is a type of noncut overA,thencanddare algebraically independent overA,namely,cdcl(A,d)andddcl(A,c).Recall from[14]that two typesp(x)andq(y)overAare weakly orthogonal if they implies a complete typer(x,y)overA.It is easy to see that if tp(c/A)and tp(d/A)are weakly orthogonal,thencanddare algebraically independent overA.We extend Marker’s result topCF environment,showing that

Theorem 3.Let K be a model of pCF,p(x)∈S1(K)a pseudo-limit type,and q(y)∈S1(K)a distance type,then p and q are weakly orthogonal.

The paper is organized as follows:For the rest of the introduction we give precise definition and preliminaries relevant to our results.

In Section 2,we study the the elementary extensions ofp-adically closed fields and their value groups.

In Section 3,we will prove Theorem 2,classifying the 1-types over an arbitrary model ofpCF.

In Section 4,we will prove Theorem 3,the orthogonality of pseudo-limit types and distance types.

1.1 Notations

LetTbe a complete theory with infinite models in a countable languageLandMa model ofT.We usually write tuples asa,b,x,yrather than,,,.ForAa subset ofM,anLA-formula is a formula with parameters fromA.Ifφ(x) is anLM-formula andA ⊆M,thenφ(A)is the collection of the realizations ofφ(x)fromA,namely,φ(A)={a ∈A|x||M|=φ(a)}.Similarly,ifX ⊆M|x|is a definable set defined by the formulaφ(x),then we useX(A)to denote the setφ(A)=X ∩A|x|.IfX ⊆Mnis definable inMandN ≻M,we sometimes useX(x)to denote the formula which definesXandX(N)the subset ofNndefined by the formulaX(x).

Assume thatA ⊆Manda ∈M.We say thatais in the algebraic closure ofA(inM),writtena ∈acl(A),if there is a formulaφ(x)overA(namely ofLA)such thatM|=φ(a),and moreover such thatφ(x)has only finitely many solutions inM.We say thatais in the definable closure ofA,a ∈dcl(A),if for someLA-formulaφ(x),ais the unique solution ofφ(x)inM.Note that both acl(-)and dcl(-)are idempotent operators,namely,acl(acl(A))=acl(A)and dcl(dcl(A))=dcl(A)for anyA ⊆M.For anyn-tuplea=(a1,...,an)∈Mn,we denote acl(A∪{a1,...,an})by acl(A,a).Similarly for dcl(A,a).

Our notation for model theory is standard,and we will assume familiarity with basic notions such as very saturated models(or monster models),skolem functions,partial types,type-definable etc.References are[11]as well as[9].

1.2 Background in p-adically Closed Fields

Letpbe a prime and Qpthe field ofp-adic numbers.We call the complete theory of Qp(in the language of rings)thetheory of p-adically closed fields,writtenpCF.Ap-adic closed field is a model ofpCF,or equivalently,a field which is elementarily equivalent to Qp.A key point is the Macintyre’s theorem[7]thatpCF has quantifier elimination in the language of rings together with new predicatesPn(x)for then-th powers for eachn ∈N+.LetKbe ap-adically closed field,we denote its multiplicative group byK∗,its valuation ring byRK,and its value group by ΓK.The value group ΓKis a model of Pr(Presburger arithmetic),namely,(ΓK,<,+)is elementary equivalent to (Z,+,<).The valuation on aKis the mapvfromKto ΓK ∪{∞}satisfying:

·∞>yandy+∞=∞+y=∞for ally ∈ΓK

·v(x)=∞x=0;

·v(x+y)≥min{v(x),v(y)}andv(x+y)=min{v(x),v(y)}whenv(x) ≠v(y);

·v(xy)=v(x)+v(y).

Note that the valuation ringRK={x ∈K|v(x)≥0}and the relationv(x)≤v(y) are definable in the language of rings (see [4]),so is quantifier-free definable in Macintyre’s language.We will freely use the variables and parameters from the value group sort.

Throughout this paper,(K,+,×,0,1) will denote a very saturated model (or monster model)ofpCF andKan(arbitrary)small elementary submodel of K,where we say that a setXissmallif |X| < |K|.We use Gmto denote the multiplicative group of K,so Gm(K)=K∗is the multiplicative group ofK.Before getting into details we recall some basic facts which will be used freely in this section and the rest of the paper.

First,the topology onKis the valuation topology.Consider the formula

wherex,yare of the home sort andγis of the value group sort.For anyc ∈Kandδ ∈ΓK,we callB(c,K,δ)an open ball of centercand radiusδ.

Fact 1.

· Thep-adic field Qpis a complete,locally compact topological field,with basis given by the setsB(c,Qp,n)forc ∈Qp,n ∈Z.

·Kis also a topological field,with basis given by the setsB(c,K,δ)forc ∈K,δ ∈ΓK,but not need to be complete or locally compact.

· For eachc ∈Kandγ ∈ΓK,B(c,K,δ)=B(c′,K,δ) wheneverc′∈B(c,K,δ).

· For eachc ∈Kandγ ∈ΓK,B(c,K,δ)is clopen.

· For eachc ∈Kandγ ∈ΓK,B(c,K,δ) is a disjoint union ofB(c0,K,δ+1),...,B(cp-1,K,δ+1)for somec0,...,cp-1∈B(c,K,δ).

It is well-known thatpCF satisfiesHensel’s Lemma:

Fact 2(Hensel).Letf(t)be a polynomial overRKin one variablet,and letα ∈RK,e ∈N.Suppose thatv(f(α))≥2e+1 andv(f′(α))≤e,wheref′denotes the derivative off.Then there exists a unique∊∈RKsuch thatv(∊)≥e+1 andf(α+∊)=0.

Recall thatPn(x)denotes the formula saying thatxis ann-th power,and thatpCF has quantifier elimination after adding predicates for allPn(x).It is easy to see from the Hensel’s Lemma that eachPn(K∗)is an open subgroup of the multiplicative groupK∗with finite index,and each coset ofPn(K∗)contains representatives from Z(see[1,7]for details).

The following lemma can also be conclude directly from the Hensel’s Lemma.Nevertheless we give a proof here for convenience.

Lemma 1.Let a,b ∈K∗.If v(a)≥v(b)+2v(n)+1,then a and a-b are in the same coset of Pn(K∗)in K∗,namely,K|=∀λ(Pn(λb)↔Pn(λ(b-a))).

Proof.Let∊∈Ksuch thatv(∊)≥2v(n)+1.Consider the polynomialf(t)=tn-(1+∊).Sincef(1)=∊andf′(1)=n.It follows from the Hensel’s Lemma thatf(t)has a root inK,which means that 1+∊is ann-th power wheneverv(∊)≥2v(n)+1.

Now suppose thatv(a)>v(b)+2v(n)+1,thenv(b-a)=min{v(a),v(b)}=v(b).Let∊=a/(b-a),we see that

so 1+∊is ann-th power.Since

we see thatbandb-aare in the same coset ofPn(K∗).□

The partial type{Pn(x)|n ∈N+}defines a subgroup of Gm,we call it thedefinable connected componentof Gm,and denote it by.Note that every coset ofis type-definable over∅.

Recall that awell-indexed sequenceinKis a sequence{ai}i∈IinKwhose termsaiare indexed by the elementsiof an infinite well-ordered set(I,<)without a last element.

Definition 1.Let{ai}i∈Ibe a well-indexed sequence inK.

· We say that{ai}i∈Iis apseudo-Cauchy sequenceif for some indexi0we have thatv(ak-aj)>v(aj-ai)wheneverk>j>i>i0.

· We say thata∗∈K is apseudo-limitof{ai}i∈Iifv(ai-a∗) is eventually strictly increasing,that is,for some indexi0,we have thatv(ak-a∗)>v(aja∗)wheneverk>j>i0.

· We say that a formulaφ(x)overKiseventually trueon{ai}i∈Iif there is somei0∈Isuch thatK|=φ(ai)for alli>i0.

Remark 1.Since K is a monster model,a compactness argument shows that any pseudo-Cauchy sequence{ai}i∈IinKhas a(not necessary unique)pseudo-limit in K.

Using cell decomposition in the form of Denef[4]or[5],the following can be easily derived,cf.[2],Lemma 4:

Fact 3.LetX ⊆Km+1be a definable subset,andbj:XKdefinable functions,forj=1,...,r.Then there exists a finite partition ofXsuch that each partAhas form

and for each(x,y)∈A,we have

withx ∈Km,D ⊆Kmdefinable,0

Remark 2.It is easy to see from Fact 3 that

· Every one-variable formulaφ(x)overKis equivalent to a disjoint disjunction of the formulas of the form

withc ∈K,γi ∈ΓK,λ ∈Z,and □ieither<,≤or no condition.

· Ifs(x)is aK-definable function anda∗∈K,then there are 0

Another goodness ofpCF is that it has definable Skolem functions(See [13]),i.e.for any formulaφ(x,y) overKwithK|=∀x∃yφ(x,y),we can find aKdefinable functionfφsuch thatK|=∀xφ(x,fφ(x)).So for anyA ⊆K,dcl(A)is an elementary substructure ofK.The reader is referred to[1]for additional details ofp-adically closed fields.

2 Extensions of Models

Lemma 2.Suppose that S is a small subset ofK,thendcl(S)=acl(S)inK.

ProofAssume thata ∈acl(S).There is a finite setDdefined overSwith the smallest cardinality such thata ∈D.Then letf(x)=∏d∈D(x-d)and Aut(K/S)the group of automorphisms of K fixingSpoint-wise.Since everyσ ∈Aut(K/S)fixesDset-wisely,we see that each coefficient offis Aut(K/S)-invariant.Thus,all coefficients offare in dcl(S)by the saturation of K.As definable Skolem functions exist,some rootboffis in dcl(dcl(S))=dcl(S).Ifb≠a,thenD{b}is defined overSwith cardinality< |D|anda ∈D{b},which is impossible.Thus,a=b ∈dcl(S).□

For anyS ⊆K,acl(S)is the same whether computed inKor K.So we have that

Corollary 1.If S ⊆K,thendcl(S)=acl(S)in K.

By[6],the algebraic closure operation acl(-)in any modelKofpCF defines apre-geometry,namely the exchange axiom is satisfied: ifa,b ∈K,A ⊆Kandb ∈acl(A,a)acl(A),thena ∈acl(A,b).Fora ∈K,we write dcl(K,a)asK〈a〉,which is an elementary extension ofK.It is easy to see from Lemma 1 and the exchange axiom that:

Lemma 3.Let a ∈KK.Then there is no proper middle extension between K ≺K〈a〉,i.e.no L such that KLK〈a〉.

Recall from[12]that Pr=Th(Z,+,<,0,{Dn}n>0)has definable Skolem functions,quantifier elimination in the language{+,<,0,{Dn}n>0},and is decidable,where eachDnis a unary predicate symbol for the set of elements divisible byn.For anyA ⊆M|=Pr,we see that dcl(A)is an elementary substructure ofM.Clearly,the value group ΓKof K is a monster model of Pr.

Lemma 4.Let K0≺K,and G an elementary substructure ofΓK extendingΓK0.Then there is K1such that K0≺K1≺K and G=ΓK1.

Proof.Let

ThenKis not empty sinceK0∈K.Applying Zorn’s Lemma to(K,⊆),and letK1be a maximal element ofK.We claim thatGis the value group ofK1.Otherwise,there isα ∈GΓK1.Take anya ∈Ksuch thatv(a)=α.Thenv(a-c)=min{v(a),v(c)} ∈Gfor eachc ∈K1.By Remark 2,for eachb ∈dcl(K1,a)=K1〈a〉,there are 0

Sov(K1〈a〉)⊆Gand thus the proper extensionK1〈a〉ofK1is also inK.A contradiction.□

We see from Lemma 4 that anyM|=Pr is isomorphic to a value group of someK≺K.In this paper,we consider any (small) model of Pr as a value group of of someK≺K.We also write dcl(M,α)asM〈α〉forα ∈ΓK.

Lemma 5.Let K′≻K and a ∈K′K such that v(a)=αΓK.ThenΓK〈a〉=ΓK〈α〉.

Proof.It is easy to see that

since ΓK ∪{α}⊆v(dcl(K,a)).Suppose for a contradiction that ΓK〈α〉is a proper subset of ΓK〈a〉.Then by Lemma 4,there isK′such thatK ≺K′≺K〈a〉 such that ΓK′=ΓK〈α〉.Since ΓK′is proper middle extension between ΓKand ΓK〈a〉,it follows thatK′is a proper middle extension betweenKandK〈a〉.This contradicts to Lemma 3.□

Corollary 2.Let M|=Prand α ∈ΓKM,then there is no middle extension between M and M〈α〉,i.e.there is no N such that MNM〈α〉.

Proof.Suppose Not,then there isβ ∈M〈α〉such that

By Lemma 4,there isK′|=pCF such that ΓK′=M.Take anya ∈K such thatv(a)=α,thenM〈α〉 is the value group ofK′〈a〉 by Lemma 5.Take anyb ∈K′〈a〉 such thatv(b)=β,then applying Lemma 5 again,M〈β〉 is the value group ofK′〈b〉.We conclude thatK′〈b〉 is a proper middle extension betweenK′andK′〈a〉.A contradiction.□

3 Classification of 1-types

Recall that K is the monster model ofpCF and Gmis the multiplicative group of K.From now on,we fixKas a small elementary submodel of K andK∗=Gm(K)the multiplicative group ofK.

Definition 2.Suppose that(Γ,+,<,0)is a model of Presburger arithmetic.We say that Λ⊆Γ is acutof Γ if

· For eachγ,β ∈Λ,ifγ ∈Λ andβ<γ,thenβ ∈Λ;

· For eachn ∈Z andγ ∈Λ,γ+n ∈Λ.

i.e.Λ is downward closed and satisfying Λ+Z=Λ.

Lemma 6.Let a∗∈KK,and

ThenΛis a cut ofΓK

Proof.It is easy to see that Λ is downward closed.It suffices to show thatδ ∈Λ impliesδ+1∈Λ.Suppose thatv(c-a∗) >δfor somec ∈Kandδ ∈ΓK,we see from Fact 1 that there arec0,...,cp-1∈Ksuch thatB(c,K,δ)is a disjoint union ofB(c0,K,δ+1),...,B(cp-1,K,δ+1).Since K is an elementary extension ofK,B(c,K,δ)is also a disjoint union ofB(c0,K,δ+1),...,B(cp-1,K,δ+1).Asa∗∈B(c,K,δ),there isiδ+1 as required.□

Theorem 4.Let K be a model of pCFandbe the definableconnected component ofGm.Then the complete1-types over K are precisely the following:

(a)The realized typestp(a/K)for each a ∈K;

(b) (distance type around a point)For each cutΛ⊆ΓK,c ∈K,and coset C of,the type pΛ,c,C saying thatΛ

(c) (Pseudo-limit type)For each pesudo-Cauchy sequence{ci}i∈I,the type(x)saying that x is a pseudo-limit of{ci}i∈I.In this case,(x)is determined by the sequence{ci}i∈I:For each formula φ(x)over K,φ(x)∈(x)iff φ(x)is eventually true on{ci}i∈I.

Proof.Letp(x)∈S1(K)be a non-realized type anda∗|=p.Let

Then Λ is a cut of ΓKby Lemma 6.Leta∗|=p.Now we have two cases:

•Case 1:There isc ∈Ksuch thatv(a∗-c)is maximal among the set{v(a∗-d):d ∈K}.Thenv(a∗-c)∈ΓKΓKrealizes the cut Λ,i.e.Λ

Claim 1.LetΣΛ,C(x,c)be the partial type saying thatΛ

Proof.Clearly,every formulaφ(x)∈p(x) is consistent with ΣΛ,C(x,c) since ΣΛ,C(x,c)⊆p.

Now suppose thatφ(x)is consistent with ΣΛ,C(x,c).We aim to show thata∗|=φ(x).By Remark 2,we can assume thatφ(x)is of the form

withα1,α2∈ΓK,d ∈K,ands ∈Z.Letb∗∈K realize the partial type

Then bothv(a∗-c)andv(b∗-c)realize the cut Λ.As bothv(a∗-c)andv(b∗-c)are not in ΓK,we have that

Ifv(c-d)∈Λ,then

which means thata∗|=α1□1v(x-d)□2α2.

Ifv(c-d)Λ,then both

realize the cut Λ,so

which also means thata∗|=α1□1v(x-d)□2α2.

We now show thata∗also realizesPn(s(x-d)).Ifv(c-d)∈Λ,we have

By Lemma 1,we see that (a∗-d),(c-d) and (b∗-d) are in the same coset ofPn(Gm).Soa∗|=Pn(s(x-d)) as required.Similarly,ifv(c-d)Λ,then we have both

Which implies thata∗-danda∗-care in the same coset ofPn(Gm),also,b∗-dandb∗-care in the same coset ofPn(Gm).Since(b∗-c)and(a∗-c)are in the same coset of,we see that K |=Pn(s(b∗-d))↔Pn(s(a∗-d)).Soa∗|=Pn(s(x-d)).This complete the proof of the Claim.□

Clearly,we see for the above Claim thatpis determined by the partial type ΣΛ,C(x,c)when Case 1 happens.

•Case 2.There is no suchcas in the previous case.First we show thatv(a∗-c)∈ΓKfor eachc ∈K.To see this,suppose that there isc ∈Ksuch thatv(a∗-c)ΓK,then for anyc≠d ∈K,we have

This contradicts our assumption.So we conclude that Λ={v(a∗-c)|c ∈K}.We claim that Λ has a well-ordered cofinal subetI.Let

Applying Zorn’s Lemma to(W,⊆),and letIbe a maximal element ofW,it is easy to see thatIis cofinal in Λ.Since Λ is a cut,it has no largest element,we see thatIis infinite.Take a sequence{ci ∈K|i ∈I}such thatv(a∗-ci)=i,Thena∗is a pseudo-limit of{ci}i∈I.

Claim 2.A formula φ(x)over K is in p(x)iff φ(x)is eventually true on{ci ∈K|i ∈I}.

Proof.Assume again that the formulaφ(x)is of the form

withα1,α2∈ΓK,d ∈Kands ∈Z.Leti0∈Isuch thatv(a∗-ci)>v(a∗-d)+2v(n)+1 for alli>i0.As

for alli>i0,we see that

Applying Lemma 1,we have that (a∗-d) and (ci-d) are in the same coset ofPn(Gm).So

for alli>i0.We conclude thatφ(x)∈piffφ(x)is eventually true on{ci ∈K|i ∈I}.This completes the proof.□

We see from Claim 1 and Claim 2 that eachp(x)∈S1(K)is either a realized type,or a distance type determined by a cut Λ,a pointc ∈Kand,a cosetCof,or a pseudo-limit type determined by a pseudo-Cauchy sequence.

Conversely,the proof of Claim 1 indicates that for each cut Λ⊆K,c ∈K,and cosetCof,the partial type ΣΛ,C(x,c) determines a complete 1-type overK.Similarly,the proof of Claim 2 indicates that each pseudo-Cauchy sequence also determines a complete 1-type overK.□

4 Orthogonality of 1-types

As we mentioned in the introduction,distance types and pseudo-limit types are the analogues of “noncut” and “cut” in theo-minimal context respectively.We aim to show the orthogonality of distance types and pseudo-limit types in this section.

Lemma 7.Let a ∈KK.Then

· tp(a/K)is a pseudo-limit type iffΓK〈a〉=ΓK.

· tp(a/K)is a distance type iffΓK〈a〉≠ΓK.Moreover,iftp(a/K)is a distance type around c ∈K thenΓK〈a〉=ΓK〈v(a-c)〉.

Proof.It is easy to see from Theorem 4 that tp(a/K) is a pseudo-limit type iffv(a-b)∈ΓKfor allb ∈K.So ΓK〈a〉=ΓKimplies that tp(a/K)is a pseudo-limit type.

Now suppose that ΓK〈a〉≠ΓK.To see that tp(a/K)is a distance type,it suffices to show thatv(a-c)ΓKfor somec ∈K.Letsbe aK-definable function such thatv(s(a))ΓK,then by Remark 2,

for some 0

For the “moreover” part,suppose that tp(a/K)is a distance type aroundc ∈K,we have seen that everyδ ∈ΓK〈a〉is of the form 1/e(v((a-d)n)+γ)with 0

Remark 3.It is easy to see from Lemma 7 that a non-realized type can not be both of distance and pseudo-limit simultaneously.

Lemma 8.Let a ∈KK and b ∈K〈a〉K,thentp(b/K)is in the same case oftp(a/K),i.e.tp(b/K)is distance type(resp.pseudo-limit type),iftp(a/K)is.

Proof.We see from Lemma 3 that thatK〈b〉=K〈a〉and hence,ΓK〈a〉=ΓK〈b〉.By Lemma 7,tp(a/K)is a distance type iff ΓK〈a〉=ΓK〈b〉≠ ΓKiff tp(b/K)is a distance type.□

Lemma 9.Iftp(c/K)is a distance type andtp(d/K)is a pseudo-limit type,then c and d are algebraic independent over K,i.e.cacl(K,d)=dcl(K,d)and dacl(K,c)=dcl(K,c).

Proof.Suppose for a contradiction thatc ∈dcl(K,d).Asc/∈K,it follows from Lemma 8 that tp(c/K)is a also a pseudo-limit type,which is impossible by Lemma 7.Similarly,we haveddcl(K,c).□

For bothaandbrealize distance types overK,we have a rough relation between tp(a/K〈b〉)and tp(b/K〈a〉):

Proposition 1.If both a and b realize distance types over K,thentp(a/K〈b〉)is in the same case oftp(b/K〈a〉),i.e.tp(a/K〈b〉)is a realized(resp.distance,pesudo-limit)type iftp(b/K〈a〉)is.

Proof.Sincea,b/∈K,we see thatb ∈K〈a〉 iffa ∈K〈b〉 by Lemma 3.So tp(a/K)is realized iff tp(b/K)is realized.

Now we assume thatb/∈K〈a〉.Suppose for a contradiction that tp(b/K〈a〉)is a distance type but tp(a/K〈b〉) is a pseudo-limit type.Then by Lemma 7 we have that ΓK〈a〉ΓK〈a,b〉and ΓK〈b〉=ΓK〈a,b〉.Sincearealizes a distance types overK,we see that ΓKΓK〈a〉,and hence ΓK〈a〉is a proper middle extension between Γ and ΓK〈b〉=ΓK〈a,b〉.Applying Lemma 7 again,there isα ∈ΓK〈b〉such that ΓK〈b〉=ΓK〈α〉.We conclude that ΓK〈a〉is a proper middle extension between Γ and ΓK〈α〉,this contradicts to Corollary 2.

Similarly,it is impossible that tp(b/K〈a〉)is pseudo-limit but tp(a/K〈b〉)is a distance type.□

We now show that pseudo-limit types and distance types are weakly orthogonal.

Proposition 2.Suppose thattp(a/K)is a distance type andtp(c/K)is a pseudo-limit of a sequence{ci}i∈I ⊆K.Thentp(c/K〈a〉)is also a pseudo-limit of the sequence{ci}i∈I.

Proof.Firstly,c/∈K〈a〉.Suppose not,K ≺K〈c〉≺K〈a〉implies thatK〈c〉=K〈a〉,whereas ΓK〈c〉=ΓKand ΓK≠ΓK〈a〉.

Secondly,tp(c/K〈a〉)can not be a distance type.Suppose not,we can assume that tp(c/K〈a〉) is a distance type around some pointf ∈K〈a〉.Iff ∈K,thenv(c-f)is maximal among{v(c-e)|e ∈K},and thus tp(c/K)is a distance type,which is a contradiction.Sof ∈K〈a〉K,and by Lemma 8,we see that tp(f/K)is a distance type around a pointd ∈K.Since

we have that

which is impossible becausev(f-d)ΓKby Lemma 7.Thus,we conclude that tp(c/K〈a〉)is a pseudo-limit of a well-indexed sequence{fj}j∈J ⊆K〈a〉.

To see that tp(c/K〈a〉)is a pseudo-limit of the sequence{ci}i∈I,it suffices to show that for eachj ∈Jthere isi ∈Isuch thatv(c-ci)≥v(c-fj).Suppose for a contradiction that there isj0∈Jsuch thatv(c-fj0)>v(c-ci)for alli ∈I.We see from Lemma 8 that tp(fj0/K)is a distance type.Suppose that tp(fj0/K)is arounde ∈K.Then,for alli ∈I,

As tp(c/K)is a pseudo-limit of{ci}i∈Iande ∈K,there isi0∈Isuch thatv(cci) >v(c-e) for alli ∈Iwithi>i0.Then we have that,for everyi ∈Iwithi>i0,

We conclude from(1)and(2)that,

and then

for alli ∈Iwithi>i0.Sincecis a pseudo-limit of{ci}i∈I,we see thatv(c-e)is maximal among{v(c-d)|d ∈K},and hence tp(c/K)is a distance type.It is a contradiction.□

We conclude the orthogonality of pseudo-limit types and distance types directly from Proposition 2:

Theorem 5.Suppose that p(x)∈S1(K)is a pseudo-limit type and q(y)∈S1(K)is a distance type,then there is r(x,y)∈S2(K)such that p(x)∪q(y)⊢r(x,y).

Proof.Take anyr(x,y)∈S2(K)such thatp(x)∪q(x)⊆r(x,y).We now show thatp(x)∪q(y)⊢r(x,y).Leta|=p(x) andc|=q(y),then it suffices to show that(a,c) |=r(x,y).Suppose that(a′,c′) |=r(x,y).Since tp(a/K)=tp(a′/K),by the saturation of K,there isc′′∈K such that tp(a,c′′/K)=tp(a′,c′/K).Sor=tp(a,c′′/K) andq=tp(c/K)=tp(c′′/K).Assume thatqis a pseudo-limit of a sequence (ci)i∈I ⊆K.We see from Proposition 2 that both tp(c/K〈a〉) and tp(c′′/K〈a〉)are pseudo-limit of the sequence(ci)i∈I.By Lemma 4,tp(c/K〈a〉)=tp(c′′/K〈a〉).So tp(a,c/K)=tp(a,c′′/K)=r(x,y)as required.□