Invariance group and invariance valuation ring of a cut

— preliminary version —

Franz-Viktor Kuhlmann

25. 3. 1998

1 Introduction

In this paper, we consider cuts in ordered abelian groups and in ordered fields. We associate to every cut an invariance group. This is the maximal convex subgroup of the ordered group (respectively, of the additive group of the ordered field) which can be added to the cut sets without changing them. This notion is a useful tool for the classification of cuts. For example, it helps to determine whether a cut is induced by the upper or lower edge of a convex subgroup, or even of a convex valuation ring.

Invariance groups or related notions have been introduced by many authors in various ways (for example, see [M], [W], [T]), and we do not claim that any of our results about cuts in ordered abelian groups (cf. Section 3) are new. We have developed these results in [K1] and applied them to derive new theorems in the valuation theory of fields, in particular in positive characteristic. These theorems can be found in [K2]. The reason for presenting the results in this paper is their application to the classification of cuts in ordered fields (cf. Section 5). Our attention was drawn to this application by a question of J. Madden. During the Special Semester in Real Algebraic Geometry and Ordered Structures, Baton Rouge 1996, he showed us the definition of the invariance valuation ring and asked for the meaning of it. This paper presents our answer to his question. It is based on the manuscript [K3].

Looking at a cut in an ordered field (K, <), one may ask whether it originates in some way from a cut in the residue field of K with respect to some real place. That is, one would like to know whether the cut can be translated into some “normal position” such that for some convex valuation ring O of (K, <) with maximal ideal M, it induces a cut in the residue field O/M via the residue map. If so, one would like to determine how this translation can be done. For instance, this can be of importance for local-global principles in real algebra. The invariance valuation ring is a key tool to answer these questions.

A remark by M. Marshall made it clear to us that some of our results of Section 5 are actually a special case of a more general setting. If v is the natural valuation of an ordered field (K, <) and O_{v} is its valuation ring, then every convex subgroup of the ordered additive group of (K, <) is an O_{v}-module. The map

M ↦ (vK \ vM, vM) ,

where vK is the value group of (K, v) and vM := {va | 0 ≠ a ∈ M}, is a bijection between the convex subgroups M of (K, <) and the cuts in the value group vK. This holds more generally for any valuation v of an arbitrary field K and the set of all O_{v}-modules M ⊆ K. We will investigate in Section 4 which information about M can be read off from the invariance group of the cut (vK \ vM, vM). We also define the invariance valuation ring of an O_{v}-module. The invariance valuation ring of a cut is then the invariance valuation ring of the invariance group of the cut.

The invariance valuation ring of a cut has been independently introduced and studied by M. Tressl in [T].

I would like to thank J. Madden, M. Marshall and S. Kuhlmann for helpful discussions. In particular, I would like to thank Jim Madden for his support and effort in Baton Rouge.

2 Preliminaries on cuts in ordered sets

Take any ordered set (S, <) (if we say “ordered”, we will always mean “totally ordered”). If S_{1}, S_{2} are nonempty subsets of S and a ∈ S, we will write a < S_{2} if a < b for all b ∈ S_{2}, and further, we will write S_{1} < S_{2} if a < S_{2} for all a ∈ S_{1}. Similarly, we use the relation ≤.

A subset S′ of S is called convex in (S, <) if for every two elements a, b ∈ S′ and every c ∈ S such that a ≤ c ≤ b, it follows that c ∈ S′. A subset Λ of S is an initial segment of S if for every a ∈ Λ and every c ∈ S with c ≤ a, it follows that c ∈ Λ. Symmetrically, Λ is a final segment of S if for every a ∈ Λ and every c ∈ S with c ≥ a, it follows that c ∈ Λ. Note that Λ is an initial segment of S if and only if Λ is convex and Λ < S \ Λ. Note that ∅ < S and S < ∅ by definition; so ∅ is an initial segment and a final segment of S.

If Λ ⊆ S and Λ′ ⊆ S are such that Λ ≤ Λ′ and S = Λ ∪ Λ′, then we will call (Λ, Λ′) a quasi-cut in S. Then Λ is an initial segment in S, Λ′ is a final segment in S, and the intersection of Λ and Λ′ consists of at most one element. If this intersection is empty, then (Λ, Λ′) will be called a cut in S. A cut (Λ, Λ′) with Λ ≠ ∅ and Λ′ ≠ ∅ will be called a Dedekind cut. If (T, <) is an extension of (S, <) and a ∈ T is such that Λ ≤ a ≤ Λ′, then we will say that a realizes (Λ, Λ′) (in (T, <)).

If a, b, c ∈ S such that a < c < b, then we will say that c is strictly between a and b. We will say that (S, <) is dense if for every two distinct elements of S there is a third element of S strictly between them. This holds if and only if there are no cuts (Λ, Λ′) in S for which Λ has a last element and Λ′ has a first element. If there is no element strictly between a and b, then b is called the immediate successor of a, and a is called the immediate predeccessor of b. Further, (S, <) is called discretely ordered if for every a ∈ S there is an immediate successor if a is not the first element in S, and an immediate predeccessor if a is not the last element in S. The properties “dense” and “discretely ordered” are mutually exclusive.

3 Cuts in ordered abelian groups

Throughout this section, let (Λ, Λ′) be a cut in the ordered abelian group (G, <). Further, H will always denote an arbitrary convex subgroup of G. Then G/H is again an ordered abelian group, with the ordering induced through the canonical epimorphism G → G/H. That is, this epimorphism preserves the relation ≤. It follows that Λ/H ≤ Λ′/H (where Λ/H denotes the image of Λ under the canonical epimorphism). That is, (Λ/H , Λ′/H) is a quasicut in G/H.

For the image of an element g ∈ G under the canonical epimorphism, i.e., the coset g + H, we prefer to write g/H since we will also have to deal with subsets of the form g + H in G. With this notation, Λ/H = {g/H | g ∈ Λ}. Recall that g + H = {g + h | h ∈ H}. We set Λ + H := {g + h | g ∈ Λ , h ∈ H}.

Note that for every g ∈ G, the addition map a ↦ a + g is an order preserving bijection from G onto G. Hence, (Λ + g , Λ′ + g) is again a cut in (G, <).

3.1 The invariance group of a cut

Theorem 3.1 The set

I(Λ) := {g ∈ G | Λ + g = Λ}

is a convex subgroup of (G, <).

Proof: It is straightforward to prove that I(Λ) is a subgroup of G. We show that I(Λ) is convex in G. Since Λ is an initial segment of G, we have that Λ + a ⊆ Λ + b whenever a ≤ b. Hence if a, b ∈ I(Λ) and g ∈ G with a ≤ g ≤ b, then

Λ = Λ + a ⊆ Λ + g ⊆ Λ + b = Λ

which shows that Λ + g = Λ, i.e. g ∈ I(Λ). This proves that I(Λ) is convex in G. □

The convex subgroup I(Λ) of (G, <) will be called the invariance group of the cut (Λ, Λ′) (or simply of Λ) in (G, <). Wherever convenient, we will abbreviate it by I.

Lemma 3.2 The group I is the largest of all convex subgroups H of G satisfying that (Λ + H) ∩ Λ′ = ∅. Further,

I = I(Λ′) := {g ∈ G | Λ′ + g = Λ′} .  (1)

Proof: Since every convex subgroup H of G contains 0, it satisfies Λ ⊆ Λ + H. The latter is an initial segment of G since it is the union of initial segments of the form Λ + g, with g ∈ H. Therefore, as G is the disjoint union of Λ and Λ′, we have that Λ = Λ + H if and only if (Λ + H) ∩ Λ′ = ∅. This proves the first assertion. The second follows from the fact that Λ + g = Λ if and only if Λ′ + g = Λ′. □

Note that if Λ admits a last or Λ′ admits a first element, then I is trivial (i.e., I = {0}). The converse is not true. Indeed, every Dedekind cut in ℚ has trivial invariance group.

Remark 3.3 Recall that an ordered abelian group is discretely ordered if and only if it has a smallest positive element. This already holds if there is at least one element in the group which has an immediate successor. Observe that if (Λ, Λ′) is a cut in a discretely ordered abelian group (G, <) with invariance group I = {0}, then Λ has a last and Λ′ has a first element. Indeed, if I = {0} and if g_{0} is the smallest positive element of G, then there is some element g ∈ Λ such that g + g_{0} ∈ Λ′. But as g + g_{0} is the immediate successor of g, we find that g is the last element of Λ and g + g_{0} is the first element of Λ′.

Remark 3.4 If 0 ∈ Λ, then I ⊆ Λ because I = 0 + I ⊆ Λ + I = Λ. Symmetrically, if 0 ∈ Λ′, then I ⊆ Λ′. We have that I = G if and only if Λ or Λ′ is empty.

Theorem 3.5 The induced quasi-cut (Λ/H , Λ′/H) in G/H is a cut if and only if H ⊆ I. If I ≠ G and H ⊆ I, then (Λ/H , Λ′/H) is a Dedekind cut in G/H.

Proof: The quasi-cut (Λ/H , Λ′/H) is a cut in G/H if and only if Λ/H ∩ Λ′/H = ∅. This holds if and only if Λ + H ⊆ Λ. By definition of I, this in turn holds if and only if H ⊆ I.

If I ≠ G then by the foregoing remark, Λ and Λ′ are non-empty. Hence also Λ/H and Λ′/H are non-empty. So (Λ/H , Λ′/H) is a Dedekind cut as soon as it is a cut. □

Corollary 3.6 If H is a final segment of Λ, i.e., if Λ is the smallest initial segment of G containing H, then H = I. The same holds if H is an initial segment of Λ′, i.e., if Λ′ is the smallest final segment of G containing H.

Proof: If H is a final segment of Λ, then 0 ∈ Λ and by Remark 3.4, I ⊆ Λ, which implies that I ⊆ H. On the other hand, it follows that 0/H is the last element of Λ/H, but is not contained in Λ′/H since the coset 0 + H = H has empty intersection with Λ′. Therefore, (Λ/H , Λ′/H) is a cut in G/H. Hence by the foregoing theorem, H ⊆ I. □

Lemma 3.7 The invariance group of the cut (Λ/I , Λ′/I) in G/I is trivial. More generally, if H ⊆ I, then the invariance group of (Λ/H , Λ′/H) is I/H.

Proof: Take H ⊆ I. Then Λ + H = Λ. It follows that for every g ∈ G, we have Λ/H + g/H = Λ/H if and only if Λ + g = Λ . Hence, I(Λ/H) = I(Λ)/H. □

In view of Remark 3.3, we obtain from this lemma:

Corollary 3.8 Assume that the group G/I is not trivial and that it is discretely ordered. Then Λ/I has a last and Λ′/I has a first element in G/I.

3.2 Shifting cuts

We will now consider shifted cuts (Λ + a , Λ′ + a) obtained from (Λ, Λ′) with arbitrary a ∈ G. Since Λ + a + g = Λ + g + a = Λ + a holds if and only if Λ + g = Λ, we have, in view of (1):

Lemma 3.9 The invariance group of (Λ + a , Λ′ + a) is

I(Λ + a) = I(Λ) = I(Λ′) = I(Λ′ + a) .

We will say that the cut (Λ, Λ′) can be shifted into H if there is some a ∈ G such that

(Λ + a) ∩ H ≠ ∅ and (Λ′ + a) ∩ H ≠ ∅ ,

or in other words, ((Λ + a) ∩ H , (Λ′ + a) ∩ H) is a Dedekind cut in H. If this is the case, then the cut ((Λ + a) ∩ H , (Λ′ + a) ∩ H) in H is realized in H if and only if (Λ, Λ′) is realized in G. Note that if (Λ, Λ′) is realized in G, then it can be shifted into every non-trivial convex subgroup of G, and its invariance group is trivial. This is a special case of the next lemma.

Theorem 3.10 The cut (Λ, Λ′) can be shifted into H if and only if H contains I as a proper subgroup.

Proof: First, we show that (Λ, Λ′) cannot be shifted into I (and hence into no subgroup contained in I). Take any a ∈ G and assume that (Λ + a) ∩ I ≠ ∅. Since I is also the invariance group of (Λ + a , Λ′ + a), it follows that (Λ + a) ∩ I = ((Λ + a) ∩ I) + I = I. This implies that (Λ′ + a) ∩ I = ∅, showing that (Λ, Λ′) cannot be shifted into I.

Now assume that H properly contains I. By Theorem 3.5, the quasi-cut (Λ/H , Λ′/H) induced by (Λ, Λ′) in G/H is not a cut. Hence, we can choose some a ∈ Λ such that a/H ∈ (Λ/H) ∩ (Λ′/H). This implies that (a + H) ∩ (Λ + H) ≠ ∅ and (a + H) ∩ (Λ′ + H) ≠ ∅. This in turn yields that (Λ − a) ∩ H ≠ ∅ and (Λ′ − a) ∩ H ≠ ∅. □

3.3 Distinguished cuts

We will say that the cut (Λ, Λ′) is a weakly distinguished cut if I is nontrivial and the image of Λ in G/I admits a last element. If in addition, 0/I is this last element, then (Λ, Λ′) will be called a distinguished cut. This name is chosen since distinguished cuts correspond to distinguished pseudo Cauchy sequences as defined by Paulo Ribenboim [R], p. 105; cf. Remark 3.15 below.

Symmetrically, if the image of Λ′ in G/I admits a first element, then we will call (Λ, Λ′) a weakly counter-distinguished cut. If this first element is 0/I, then (Λ, Λ′) is called a counter-distinguished cut. Note that (Λ, Λ′) is a counter-distinguished cut if and only if (−Λ′, −Λ) is a distinguished cut, and the same holds for the “weakly” version. Observe further that if G/I is densely ordered, then (Λ, Λ′) cannot be weakly distinguished and weakly counter-distinguished at the same time.

Example 3.11 Take any convex subgroup H of G. If Λ is the smallest initial segment of G containing H, then I = H by Corollary 3.6. It follows that the image of Λ in G/H has the last element 0/H. Hence, (Λ, Λ′) is a distinguished cut if H is non-trivial.

If on the other hand, Λ′ is the smallest final segment of G containing H, then again, I = H. But now it follows that the image of Λ′ in G/H has the first element 0/H. Hence, (Λ, Λ′) is a counter-distinguished cut if H is non-trivial. If G/H is densely ordered, then it has no largest negative element. This will imply that the image of Λ in G/H has no last element. Hence in this case, (Λ, Λ′) is not a distinguished cut. ♢

Lemma 3.12 The cut (Λ, Λ′) is weakly distinguished if and only if there is some g ∈ G such that the shifted cut (Λ + g , Λ′ + g) is distinguished. Similarly, it is weakly counter-distinguished if and only if there is some g ∈ G such that (Λ + g , Λ′ + g) is counter-distinguished.

Proof: Take any g ∈ G. Then −g/I is the last element of the image of Λ/I if and only if 0/I is the last element of (Λ + g)/I in G/I. The “counter”-proof is similar. □

The next lemma tells us that the above example represented all distinguished and counter-distinguished cuts. In other words, the distinguished cuts are given by the upper edges of non-trivial convex subgroups, and the counter-distinguished cuts are given by the lower edges of non-trivial convex subgroups.

Lemma 3.13 The cut (Λ, Λ′) is distinguished if and only if I is a final segment of Λ. Symmetrically, (Λ, Λ′) is counter-distinguished if and only if I is an initial segment of Λ′.

Proof: If I is a final segment of Λ, then 0 ∈ Λ and g > I ⇒ g ∉ Λ. This implies that 0/I is the last element of Λ/I. Conversely, if 0/I is the last element of Λ/I, then there is some g ∈ Λ ∩ I. But then, I = g + I ⊆ Λ. If a ∈ G with a > I, then a/I > 0/I, hence a/I cannot be in Λ/I, and a cannot be in Λ. This proves that I is a final segment of Λ. The “counter” proof is similar. □

Corollary 3.14 The cut (Λ, Λ′) is weakly distinguished if and only if for some g ∈ G, I is a final segment of Λ + g. Symmetrically, (Λ, Λ′) is weakly counter-distinguished if and only if for some g ∈ G, I is an initial segment of Λ′ + g.

Remark 3.15 Paulo Ribenboim [R] defines distinguished pseudo Cauchy sequences to be those pseudo Cauchy sequences (a_{ρ})_{ρ<λ} in a valued field (K, v) for which the set

{v(a_{ρ+1} − a_{ρ}) | ρ < λ} = {v(a_{σ} − a_{ρ}) | ρ < σ < λ}  (2)

is cofinal in a convex subgroup H of the value group vK of K. But this just means that H is a final segment of the smallest initial segment Λ containing the set (2). Hence, the cut (Λ , vK \ Λ) determined by the set (2) is distinguished if and only if the pseudo Cauchy sequences (a_{ρ})_{ρ<λ} is.

By means of Lemma 3.13, no cut is distinguished and counter-distinguished at the same time. But we have:

Theorem 3.16 Assume that (Λ, Λ′) is a Dedekind cut in G. Then the following are equivalent:

1) (Λ, Λ′) is weakly distinguished and weakly counter-distinguished,

2) with the induced order, G/I is discretely ordered.

Proof: If (Λ, Λ′) is weakly distinguished and weakly counter-distinguished, then by Corollary 3.14 there are a, a′ ∈ G such that a + I is a final segment of Λ, and a′ + I is an initial segment of Λ′. But then, a′/I is the immediate successor of a/I in G/I. Now if any element of an ordered abelian group has an immediate successor, then the group must be discretely ordered.

For the converse, assume that G/I is discretely ordered. Since (Λ, Λ′) is a Dedekind cut in G, we know from Remark 3.4 that I ≠ G. Hence by Corollary 3.8, Λ/I has a last element and Λ′/I has a first element in G/I. This shows that (Λ, Λ′) is weakly distinguished and weakly counter-distinguished. □

Example 3.17 Take ℤ × ℚ with the lexicographic order, and set

Λ := {(m, q) | m ≤ 0 , q ∈ ℚ} and Λ′ := {(m, q) | m ≥ 1 , q ∈ ℚ} .

Then (Λ, Λ′) is a cut with invariance group I = {0}×ℚ. We have the canonical isomorphism (ℤ×ℚ)/I ≃ ℤ, which sends the cut induced by (Λ, Λ′) to the cut ({m ∈ ℤ | m ≤ 0} , {m ∈ ℤ | m ≥ 1}) in ℤ. Hence, (Λ, Λ′) is distinguished and weakly counter-distinguished. Further, {0} × ℚ = (0, 0) + {0} × ℚ is a final segment of Λ, and {1} × ℚ = (1, 0) + {0} × ℚ is an initial segment of Λ′. The cut (Λ − (1, 0) , Λ′ − (1, 0)) in ℤ × ℚ is weakly distinguished and counter-distinguished. ♢

4 Valuation rings and their modules in a field

Take any valued field (K, v). Denote the valuation ring of v by O_{v}. Then the set R of all valuation rings which contain O_{v} is linearly ordered by inclusion. For every O ∈ R, we set

H(O) := vO ∩ −vO = vO^{×} .

This is a convex subgroup of the value group vK. In fact, v is finer or equal to the valuation w associated with O, the value group of w is canonically isomorphic to vK/H(O), and the value group of the valuation induced by v on the residue field Kw is canonically isomorphic to H(O).

Conversely, for every convex subgroup H of vK, we set

O(H) := {b ∈ K | ∃α ∈ H : α ≤ vb} .  (3)

That is, vO(H) is the smallest final segment of vK containing H. Note that

O(H)^{×} = v^{−1}(H) .

We recall the following fact from general valuation theory:

Lemma 4.1 The map H ↦ O(H) is an order preserving bijection from the set of all convex subgroups of vK onto the set R of all valuation rings which contain O_{v}. Its inverse is the map R ∋ O ↦ H(O).

4.1 O_{v}-modules in K

Now we consider the O_{v}-modules M ⊂ K. For every such module M, we have that

vM := {va | 0 ≠ a ∈ M}

is a final segment of vK, and that

M = {a ∈ K | va ∈ vM} .  (4)

Hence:

Lemma 4.2 The map v establishes a bijection between the O_{v}-modules M ⊆ K and the final segments of vK. The map M ↦ (vK \ vM, vM) is a bijection between the O_{v}-modules M ⊆ K and the cuts in the value group vK.

Therefore, we can expect to read off information about M from the invariance group of the associated final segment. We start with the case of M = O ∈ R. Note that for every O ∈ R with maximal ideal M, we have that O and M are O_{v}-modules.

Lemma 4.3 For every O ∈ R with maximal ideal M,

H(O) = I(vO) = I(vM) and O = O(I(vO)) = O(I(vM)) .  (5)

Proof: Since O is the disjoint union of O^{×} and M, (4) shows that vO is the disjoint union of vO^{×} = H(O) and vM. Since the latter is a final segment of vK, it follows that H(O) < vM. Thus, H(O) is an initial segment of vO and a final segment of vK \ vM. Now (5) follows from Corollary 3.6 and Lemma 4.1. □

The next lemma shows that there is at most one O ∈ R such that cM = O or cM = M for some non-zero c ∈ K.

Lemma 4.4 Take any O_{v}-module M ⊂ K, and O ∈ R with maximal ideal M. If for some c ∈ K, c ≠ 0, we have that cM = O or that cM = M, then O = O(I(vM)).

Proof: By Lemma 4.2, cM = O is equivalent to vcM = vO, and cM = M is equivalent to vcM = vM. Both cases imply that

I(vM) = I(vc + vM) = I(vcM) = I(vO) = I(vM) = H(O) ,

where we have used Lemma 3.9 for the first and (5) for the last two equalities. By Lemma 4.1, it follows that O = O(H(O)) = O(I(vM)). □

From this lemma together with (5), we deduce:

Corollary 4.5 For i = 1, 2, take O_{i} ∈ R with maximal ideal M_{i}. If for some c ∈ K, c ≠ 0, we have that cO_{1} = O_{2} or cO_{1} = M_{2} or cM_{1} = O_{2} or cM_{1} = M_{2} , then O_{1} = O_{2} .

For the sake of completeness, let us add:

Lemma 4.6 Take O ∈ R with maximal ideal M. Then there is some nonzero c ∈ K such that cO = M if and only if vK/H(O) is discretely ordered.

Proof: For an arbitrary valuation ring, it holds that its maximal ideal is generated by one element c if and only if the value of c is the smallest positive element in the associated value group. As vK/H(O) is the value group associated to O, this yields our assertion. □

For later use, we prove:

Lemma 4.7 Take any O_{v}-module M ⊂ K. Then O(I(vM)) is the largest of all rings O ∈ R with the property that M is an O-module.

Proof: We have that M is an O-module if and only if OM = M, which by Lemma 4.2 is equivalent to vO + vM = vOM = vM. As vM is a final segment of vK and H(O) = vO^{×} is an initial segment of vO, the latter is equivalent to H(O) + vM = vM, which in turn holds if and only if H(O) ⊆ I(vM). By means of Lemma 4.1, this is equivalent to O ⊆ O(I(vM)). □

4.2 The invariance valuation ring of an O_{v}-module in K

Take an O_{v}-module M ⊂ K. We set

O(M) := {b ∈ K | bM ⊆ M} and M(M) = {b ∈ K | bM ⊂_{≠} M} .  (6)

Example 4.8 For every O ∈ R with maximal ideal M,

O(O) = O(M) = O and M(O) = M(M) = M .
♢

Theorem 4.9 For every O_{v}-module M ⊂ K, O(M) is a valuation ring of K with maximal ideal M(M) and containing O_{v}. It is the largest of all valuation rings O of K for which M is an O-module. Further, O(M)^{×} = {b ∈ K | bM = M}.

Proof: It is straightforward to prove that O(M) is a ring. As M is an O_{v}-module, we have that O_{v}M = M. Hence, O_{v} ⊆ O(M). By general valuation theory, it follows that O(M) is a valuation ring.

The inclusion {b ∈ K | bM = M} ⊆ O(M)^{×} holds since bM = M ⇔ M = b^{−1}M for b ≠ 0. The converse inclusion holds since if b, b^{−1} ∈ O(M), then M = bb^{−1}M ⊆ bM ⊆ M and thus, bM = M.

In every valuation ring, the unique maximal ideal consists precisely of all non-units. Hence by what we have already proved, the maximal ideal of O(M) is O(M) \ {b ∈ K | bM = M} = M(M).

Finally, it follows directly from the definition that if O is a valuation ring of K such that M is an O-module, then O ⊆ O(M). On the other hand, it is also follows from the definition that M is a O(M)- module; therefore, O(M) is the largest of all valuation rings of K with this property. □

By this theorem and Lemma 4.7 we find that

O(M) = O(I(vM)) ,

which shows that definition (6) is coherent with definition (3). Now Lemma 4.1 shows that H(O(M)) = I(vM). In other words:

Lemma 4.10 The group

H(M) := H(O(M))

is the invariance group of vM in vK.

Lemma 4.11 For every nonzero c ∈ K,

O(cM) = O(M) .

Proof: Since c is invertible, we have that cbM = bcM ⊆ cM if and only if bM ⊆ M. □

Lemma 4.12 1) For every nonzero c ∈ K,

either cM ⊆ M(M) , or O(M) ⊆ cM .  (7)

2) Take O ∈ R and denote its maximal ideal by M. If O(M) ⊂_{≠} O, then there is some c ∈ K, c ≠ 0, such that

M ⊂_{≠} cM ⊂_{≠} O .  (8)

Proof: To start with, we note that for every c ∈ K, c ≠ 0, and every convex valuation ring O of K,

(vK \ vcM) ∩ H(O) ≠ ∅ ⇔ cM ⊂_{≠} O and vcM ∩ H(O) ≠ ∅ ⇔ M ⊂_{≠} cM .  (9)

To see this, recall that H(O) = vO^{×} and observe that cM is an O_{v}-module in K. Hence, K \ cM ∩ O^{×} ≠ ∅ ⇔ cM ⊂_{≠} O, and cM ∩ O^{×} ≠ ∅ ⇔ M ⊂_{≠} cM. Now (9) follows by means of Lemma 4.1.

1): We take O = O(M). Then H(O) is the invariance group of vcM. Hence, vcM ∩ H(O) ≠ ∅ implies that H(O) ⊆ vcM, which in turn implies that (vK \ vcM) ∩ H(O) = ∅. By means of (9), this yields our assertion. 2): From O(M) ⊂_{≠} O it follows that H(M) ⊂_{≠} H(O). This in turn implies by Lemma 4.10 Theorem 3.10 that the cut (vK \ vM, vM) can be shifted into H(O). That is, there is some c ∈ K such that

((vK \ vM) + vc) ∩ H(O) ≠ ∅ and (vM + vc) ∩ H(O) ≠ ∅ .

By (9), this implies (8). □

Lemma 4.13 Take any O_{v}-module M ⊂ K. Then cM = M(M) holds for some nonzero c ∈ K if and only if the cut (vK \ vM) , vM) is weakly distinguished. Similarly, cM = O(M) holds for some nonzero c ∈ K if and only if (vK \ vM) , vM) is weakly counter-distinguished.

Proof: We observe that (H(M), vM(M)) = (vO(M)^{×}, vM(M)) is a cut in the ordered set vO(M). By Lemma 4.2, cM = M(M) if and only if vcM = vM(M), which in turn holds if and only if (vK \ vcM) ∩ vO(M) = H(M). But this is true if and only if H(M) is a final segment of vK \ vcM. Now this holds if and only if (vK \ vcM)/H(M) admits 0/H(M) as its last elements, or in other words, if and only if the cut (vK \ vcM , vcM) is distinguished. As (vK \ vcM , vcM) = ((vK \ vM) + vc , vM + vc), Lemma 3.12 shows that this holds if and only if the cut (vK \ vM , vM) is weakly distinguished. The “counter”-proof is similar. □

5 Cuts in ordered fields

Throughout this section, let (Λ, Λ′) be a cut in the ordered field (K, <). For every c ∈ K, c > 0, the multiplication b ↦ cb is an order preserving bijection. Hence for every such c and every a ∈ K,

(cΛ + a , cΛ′ + a)

is again a cut in (K, <). For c < 0, the multiplication b ↦ cb is an order reversing bijection. Hence for every such c and every a ∈ K,

(cΛ′ + a , cΛ + a)

is again a cut in (K, <). We have:

Lemma 5.1 For every nonzero c ∈ K and all a ∈ K,

I(cΛ + a) = cI(Λ) .

Proof: We know from Lemma 3.9 that I(cΛ + a) = I(cΛ). Since c is invertible, we have that bcΛ = cΛ ⇔ cbΛ = cΛ ⇔ bΛ = Λ. This proves that I(cΛ) = cI(Λ). □

On the basis of this lemma, we can prove:

Lemma 5.2 Take a, c ∈ K, c > 0. Then (Λ, Λ′) is weakly distinguished if and only if (cΛ + a , cΛ′ + a) is. The same holds for “weakly counter-distinguished” in the place of “weakly distinguished”.

Proof: The reader may verify: g/I(Λ) is the last element of Λ/I(Λ) if and only if cg + a is the last element of (cΛ + a)/cI(Λ). Since cI(Λ) = I(cΛ + a) by the foregoing lemma, this gives our assertion. Similarly for “weakly counter-distinguished”. □

For the case of c < 0, one shows similarly that (Λ, Λ′) is weakly distinguished if and only if (cΛ′+a , cΛ+a) is weakly counter-distinguished.

For the rest of this paper, we will denote by v the natural valuation of (K, <). It is characterized by the fact that its residue field O_{v}/M_{v} is an archimedean ordered field (hence embeddable in ℝ). By definition of the natural valuation, va = vb holds if and only if a and b are archimedean equivalent elements of (K, <), and

va ≥ vb ⇔ ∃n ∈ ℕ : n|b| ≥ |a| .  (10)

Therefore,

Lemma 5.3 The convex subgroups of the ordered additive group of (K, <) are precisely the O_{v}-submodules of K.

This lemma also shows that every subring of K containing O_{v} is convex. On the other hand, a convex valuation ring of (K, <), being an O_{v}-module and containing 1, must contain O_{v}. Therefore, the set of all convex valuation rings of (K, <) is precisely the set R of all valuation rings which contain O_{v}.

The reader should note that already on every ordered abelian group, (10) defines a group valuation. The facts of the next section carry over literally to this situation. This is important since if one considers orders on non-commutative rings, the underlying additive group is still an “ordinary” ordered abelian group. For further basic properties of the natural valuation, see [KS] or [K2].

5.1 A valuation theoretic interpretation of the invariance group

We will now give an interpretation of the invariance group in terms of the natural valuation. We set

v(Λ′ − Λ) := {v(b − a) | a ∈ Λ , b ∈ Λ′} .

Lemma 5.4 For every cut (Λ, Λ′) in (K, <),

vI(Λ) = vK \ v(Λ′ − Λ) .

Proof: It follows directly from the definition that I(Λ) = K \ {±(b − a) | a ∈ Λ , b ∈ Λ′}. Hence, our assertion follows from Lemma 4.2. □

If (L, <) is an ordered field extending (K, <), then we will also write v for the natural valuation of (L, <). Then (L|K, v) is an extension of valued fields. For any x ∈ L (which we will later assume to realize the cut (Λ, Λ′)), we consider the value set

v(x − K) := {v(x − b) | b ∈ K} ⊆ vK(x) .

Lemma 5.5 a) If v(x − K) contains an element α ∉ vK, then α is the largest element of v(x − K).

b) The set v(x − K) ∩ vK is an initial segment of vK.

Proof: Take α, β ∈ v(x − K) such that α > β, and a, b ∈ K such that α = v(x − a), β = v(x − b). Then β = min{v(x − a), v(x − b)} = v(a − b) ∈ vK. This proves a). Now take α ∈ v(x − K), a ∈ K such that α = v(x − a), and β ∈ vK such that α > β. Choose c ∈ K such that vc = β. Then with b := a + c, we have that β = min{α, β} = min{v(x − a), vc} = v(x − b) ∈ v(x − K). This proves b). □

Lemma 5.6 If the cut (Λ, Λ′) in (K, <) is realized by an element x in an extension (L, <), then

vI(Λ) = vK \ v(x − K) = {α ∈ vK | α > v(x − b) for all b ∈ K} .  (11)

Proof: The second equation is an easy consequence of the foregoing lemma. The first equation follows from Lemma 5.4 if we show that

v(x − K) ∩ vK = v(Λ′ − Λ) .

Take any a ∈ Λ , b ∈ Λ′. Since x − a ≤ b − a and b − x ≤ b − a, we have that v(x − a) ≥ v(b − a) and v(b − x) ≥ v(b − a). If v(b − a) = v(x − a) or v(b − a) = v(b − x) = v(x − b), then v(b − a) ∈ v(x − K). If v(x − a) > v(b − a), then v(b − a) = min{v(b − a), v(x − a)} = v(x − b) ∈ v(x − K). If v(b − x) > v(b − a), then v(b − a) = min{v(b − a), v(b − x)} = v(x − a) ∈ v(x − K). Hence in all cases, v(b − a) ∈ v(x − K). This proves that v(Λ′ − Λ) ⊆ v(x − K) ∩ vK.

To prove the converse inclusion, take b ∈ K such that v(x − b) ∈ vK. Take c ∈ K such that c > 0 and vc = v(x − b). Then there is some positive integer n such that nc > |x − b|. If b ∈ Λ, then this implies that b + nc > x and thus, b + nc ∈ Λ′. Then v(x − b) = vc = vnc = v((b + nc) − b) ∈ v(Λ′ − Λ). If b ∈ Λ′, then b − nc < x, hence b − nc ∈ Λ and again, v(x − b) = vc = vnc = v(b − (b − nc)) ∈ v(Λ′ − Λ). □

We given examples to show how to apply this lemma.

Example 5.7 Assume that t is algebraically independent over ℚ. Take v to be the t-adic valuation on ℚ(t), that is, vt > 0. The associated place, which sends t to 0, is real since its residue field is ℚ. Hence there is an order < on ℚ(t) such that v is the natural valuation of (ℚ(t), <). The field ℝ(t) carries a canonical extension of v, the t-adic valuation on ℝ(t), and a corresponding extension of <. Note that vr = 0 for every nonzero r in the subfield ℝ of ℝ(t).

Now take any r ∈ ℝ \ ℚ. Viewing r as an element of the extension field (ℝ(t), <) of (ℚ(t), <), we take (Λ, Λ′) to be the cut induced by r in (ℚ(t), <). Take any b ∈ ℚ(t). If vb < 0 = vr, then v(r −b) = vb < 0. If vb ≥ 0, then the residue \overline{b} := b + M_{v} of b lies in ℚ. On the other hand, the residue of r is r itself, not lying in ℚ. Hence, \overline{r − b} = \overline{r} − \overline{b} ≠ 0, which implies that v(r − b) = 0. We have thus shown that 0 is the maximal element of v(r − ℚ(t)). Hence by the foregoing lemma, vI(Λ) = {α ∈ vK | α > 0}. But the latter is the same as vM_{v}. Hence by Lemma 4.2,

I(Λ) = M_{v} .

In other words: precisely the infinitesimals leave the cut (Λ, Λ′) invariant.

We analyze the corresponding cut (vK \ vI(Λ) , vI(Λ)) in vK. We observe that vK \ vI(Λ) = {α ∈ vK | α ≤ 0} has last element 0. We conclude that the invariance group of (vK \ vI(Λ) , vI(Λ)) is {0} and that the cut is distinguished. ♢

Example 5.8 We build on the last example. Now we take K to be the real closure of ℚ(t). Then vK = ℚ. Take any s ∈ ℝ \ ℚ. We set Λ := {d ∈ K | d ≤ 0} ∪ {b ∈ K | vb > s} and Λ′ := {b ∈ K | vb < s}. Then (Λ, Λ′) is a cut in (K, <). The value (with respect to the natural valuation) of any element realizing the cut in some extension field will realize the cut ({α ∈ ℚ | α < r} , {α ∈ ℚ | α > r}) in the value group of the extension field. By Lemma 5.6, vI(Λ) = {α ∈ ℚ | α > r}. The corresponding cut (vK\vI(Λ) , vI(Λ)) in vK is just the above cut induced by s; its invariance group is {0} and it is neither weakly distinguished nor weakly counter-distinguished. ♢

If K and L have the same value group and the same residue field (more precisely, if the canonical embeddings of vK in vL and of Kv in Lv are onto), then the extension (L|K, v) is called immediate.

In this case, every element x ∈ L is the pseudo limit of a pseudo Cauchy sequence (a_{ν})_{ν<λ} in (K, v); cf. [R]. If x ∉ K, then this pseudo Cauchy sequence can be chosen such that it has no pseudo limit in K. The breadth of (a_{ν})_{ν<λ} is defined to be the O_{v}-ideal

{a ∈ K | va > v(a_{ν+1} − a_{ν}) for all ν < λ} .

In fact, this is the set of all elements a ∈ K with the property that x is a pseudo limit of (a_{ν})_{ν<λ} if and only if x + a is.

Theorem 5.9 Suppose that the cut (Λ, Λ′) is not realized in (K, <), but is realized by an element x in an extension (L, <) such that (L|K, v) is immediate. If (a_{ν})_{ν<λ} is a pseudo Cauchy sequence in (K, v) having pseudo limit x but not having a pseudo limit in K, then its breadth is equal to I(Λ).

Proof: Let us denote by I the breadth of (a_{ν})_{ν<λ}. Then vI = {α ∈ vK | α > v(a_{ν+1}−a_{ν}) for all ν < λ}. By Lemma 4.2 and Lemma 5.6, it suffices to show that vI = {α ∈ vK | α > v(x − b) for all b ∈ K}. As x is a pseudo limit of (a_{ν})_{ν<λ}, we have by definition that v(a_{ν+1} − a_{ν}) = v(x − a_{ν}) ∈ v(x − K) for all ν < λ. This shows that

vI(Λ) = {α ∈ K | α > v(x − b) for all b ∈ K} ⊆ {α ∈ K | α > v(a_{ν+1} − a_{ν}) for all ν < λ} = vI .

To prove the converse inclusion, we take any b ∈ K. Suppose that v(x − b) > v(a_{ν+1} − a_{ν}) for all ν < λ. But then, since x is a pseudo limit, we have that v(a_{ν+1} − a_{ν}) = min{v(a_{ν+1} − a_{ν}), v(x − b)} = min{v(x − a_{ν}), v(x − b)} = v(b − a_{ν}). This shows that b ∈ K is also a pseudo limit of our pseudo Cauchy sequence, in contradiction to our assumption that there be no pseudo limit in K. This proves that the sequence v(a_{ν+1} − a_{ν}), ν < λ, is cofinal in v(x − K), which implies that the inclusion “⊇” holds. □

The completion K^{c} of (K, v) (with respect to the topology induced by v) admits a canonical extension of v and of the order from K. With this extension of v, (K^{c}|K, v) is an immediate extension, and the breadth of every pseudo Cauchy sequence without a pseudo limit in K is {0}. Thus, we obtain the following

Corollary 5.10 If the cut (Λ, Λ′) is realized in the completion of (K, v), then I(Λ) = {0}.

Our theorem can be used to construct interesting examples. Here is one.

Example 5.11 Assume that the elements t_{i}, i ∈ ℕ, are algebraically independent over ℚ. Define a valuation on K := ℚ(t_{i} | i ∈ ℕ) by setting vt_{1} ≫ vt_{2} ≫ … ≫ vt_{i} ≫ … ≫ 0. That is,

vK = ℤvt_{1} ⊕ ℤvt_{2} ⊕ … ⊕ ℤvt_{i} ⊕ …,

lexicographically ordered. Now define a pseudo Cauchy sequence in (K, v) by setting a_{ν} := ∑_{i=1}^{ν} t_{i}^{−1} for all ν ∈ ℕ. This is a pseudo Cauchy sequence since v(a_{ν+1} − a_{ν}) = vt_{ν+1}^{−1} = −vt_{ν+1} is monotonically increasing with ν. Observe that this sequence of values is cofinal in the negative part of the value group vK. Hence, the breadth of our pseudo Cauchy sequence is just O_{v}. The pseudo Cauchy sequence has no pseudo limit in K (it can actually be shown using Kaplansky’s theory that every such pseudo limit must be transcendental over K).

The place assiociated with v is real. Indeed, it is easy to show that O_{v}/M_{v} = ℚ. Therefore, there exists a order < on K such that v is the natural valuation of (K, <). Now we consider the cut (Λ, Λ′) induced in (K, <) by any pseudo limit of our pseudo Cauchy sequence in any extension field. Then by Theorem 5.9,

I(Λ) = O_{v} .

The reader should note that vK has no smallest nonzero convex subgroup. By Lemma 4.1, this means that there is no smallest valuation ring properly containing O_{v}.

In contrast to Example 5.7, we now have that vI(Λ) = vO_{v} = {α ∈ vK | α ≥ 0} has first element 0. We conclude that the invariance group of (vK \ vI(Λ) , vI(Λ)) is again {0} and that this time, the cut is counter-distinguished. As vK obviously has no smallest positive element, it is not discretely ordered. Hence by Theorem 3.16, the cut (vK \ vI(Λ) , vI(Λ)) cannot be weakly distinguished. ♢

5.2 The invariance valuation ring of a cut

We set

O(Λ) := O(I(Λ)) = {b ∈ K | bI(Λ) ⊆ I(Λ)}

and denote the maximal ideal O(I(Λ)) of O(Λ) by M(Λ). Note that since I(Λ) = I(Λ′),

O(Λ) = {b ∈ K | bI(Λ′) ⊆ I(Λ′)} =: O(Λ′) .

From Theorem 4.9 and Lemma 4.10, we obtain:

Corollary 5.12 The ring O(Λ) is a convex valuation ring of (K, <). The group

H(Λ) := H(O(Λ))

is the invariance group of vI(Λ) in vK. Further, H(Λ′) := H(O(Λ′)) = H(O(Λ)) = H(Λ).

The convex valuation ring O(Λ) of (K, <) will be called the invariance valuation ring of the cut (Λ, Λ′) (or simply of Λ) in (K, <). From Lemma 5.1 and Lemma 4.11, we obtain:

Lemma 5.13 For all a, c ∈ K, c ≠ 0, we have that

O(cΛ + a) = O(Λ) and H(cΛ + a) = H(Λ) .

5.3 Projecting cuts into residue fields

Take a convex valuation ring O of (K, <) with maximal ideal M. Its residue field O/M is again an ordered field, with the order induced through the residue map. We will say that the cut (Λ, Λ′) can be projected into the residue field O/M if there are elements a, c ∈ K such that c > 0 and that (cΛ + a , cΛ′ + a) induces a Dedekind cut in O/M via the residue map. This means that

( ((cΛ + a) ∩ O)/M , ((cΛ′ + a) ∩ O)/M)  (12)

is a cut in O/M, with both sets nonempty.

Lemma 5.14 The cut (Λ, Λ′) can be projected into the residue field O/M if and only if there is some c ∈ K such that

M ⊆ cI(Λ) ⊂_{≠} O .  (13)

Proof: Take 0 ≠ c ∈ K. By Theorem 3.10, the cut (cΛ , cΛ′) can be shifted into O if and only if cI(Λ) = I(cΛ) ⊂_{≠} O. From Theorem 3.5 we know that (12) is a cut (and hence a Dedekind cut) in O/M if and only if M ⊆ I(cΛ) = cI(Λ). □

From this lemma together with part 2) of Lemma 4.12, we obtain that (Λ, Λ′) can be projected into O/M if O(Λ) ⊂_{≠} O. On the other hand, we have:

Lemma 5.15 If O ⊂_{≠} O(Λ), then (Λ, Λ′) cannot be projected into O/M.

Proof: Assume that O ⊂_{≠} O(Λ). Then by general valuation theory, M(Λ) ⊂_{≠} M. If cI(Λ) ⊂_{≠} O for some c ∈ K, then cI(Λ) ⊂_{≠} O(Λ). By (7), this implies that cI(Λ) ⊆ M(Λ) ⊂_{≠} M. Hence by Lemma 5.14, (Λ, Λ′) cannot be projected into O/M. □

Now it remains to determine under which conditions (Λ, Λ′) can be projected into the residue field O(Λ)/M(Λ). By Lemma 5.14, this is the case if and only if there is c ∈ K such that M(Λ) ⊆ cI(Λ) ⊂_{≠} O(Λ). Since O(Λ) = O(I(Λ)) and M(Λ) = M(I(Λ)), (7) shows that this holds if and only if M(Λ) = cI(Λ). By Lemma 4.13, this in turn holds if and only if the cut (vK \ vI(Λ) , vI(Λ)) is weakly distinguished. We summarize what we have proved:

Theorem 5.16 1) Take any convex valuation ring O of (K, <). If O(Λ) ⊂_{≠} O, then the cut (Λ, Λ′) can be projected into the residue field O/M. If O ⊂_{≠} O(Λ), then it cannot be projected into O/M.

2) The cut (Λ, Λ′) can be projected into O(Λ)/M(Λ) if and only if the cut (vK \vI(Λ) , vI(Λ)) is weakly distinguished.

Example 5.17 Consider our Example 5.7. We have that I(Λ) = M_{v} and therefore, O(Λ) = O(M_{v}) = O_{v}. We have already seen that the cut (vK \ vI(Λ) , vI(Λ)) is distinguished. Hence by our theorem, the cut (Λ, Λ′) can be projected into O_{v}/M_{v} = ℚ. This is what we expected since by construction, the cut comes from the subfield ℚ. In fact, for the projection we can set c = 1 and a = 0.

Now consider our Example 5.8. We have noted that the invariance group of (vK \ vI(Λ) , vI(Λ)) is {0}. Hence by Corollary 5.12, H(Λ) = {0} and consequently, O(Λ) = O(H(Λ)) = O({0}) = O_{v}, as before. But now, the cut (vK \ vI(Λ) , vI(Λ)) is neither weakly distinguished nor weakly counterdistinguished. Hence by our theorem, the cut (Λ, Λ′) cannot be projected into O_{v}/M_{v} (which in this case is the real closure of ℚ). This is because the cut comes from a cut in the value group.

Finally, consider our Example 5.11. We have that I(Λ) = O_{v} and therefore again, O(Λ) = O(M_{v}) = O_{v}. We noted already that the cut (vK \ vI(Λ) , vI(Λ)) is counter-distinguished but not weakly distinguished. Hence by our theorem, the cut (Λ, Λ′) cannot be projected into O_{v}/M_{v} = ℚ. Even worse: by our construction, there is no smallest convex valuation ring O of (K, <) such that (Λ, Λ′) can be projected into O/M.

The convex valuation rings of (K, <) correspond bijectively to the convex subgroups of vK = ℤvt_{1} ⊕ ℤvt_{2} ⊕ … ⊕ ℤvt_{i} ⊕ . . .. These convex subgroups are precisely {0} and the subgroups of the form H_{k} := ℤvt_{k} ⊕ ℤvt_{k+1} ⊕ …, for all k ∈ ℕ. We set O_{k} := O(H_{k}) and denote its maximal ideal by M_{k}. Then the residue field O_{k}/M_{k} can be identified with the field ℚ(t_{i} | i ≥ k). Looking at the pseudo Cauchy sequence defined in Example 5.11, we see that precisely the summands t_{1}^{−1}, …, t_{k−1}^{−1} of our a_{ν}’s do not lie in O_{k}. So if we define a new pseudo Cauchy sequence by setting a′_{ν} := a_{ν+k−1} − a_{k−1}, then we obtain a sequence consisting of elements in O_{k}, which consequently also defines a cut in O_{k}. In fact, this new cut is (Λ − a_{k−1} , Λ′ − a_{k−1}). The projection of the new pseudo Cauchy sequence (a′_{ν})_{ν∈ℕ} via the residue map into the residue field O_{k}/M_{k} = ℚ(t_{i} | i ≥ k) renders a pseudo Cauchy sequence without a limit in that field. Actually, since we regard the residue field ℚ(t_{i} | i ≥ k) as a subfield of K, the projection is the identity on the elements of the pseudo Cauchy sequence, and the cut in the residue field is contained by cutting dowm (Λ−a_{k−1} , Λ′ −a_{k−1}) . This cut is again a Dedekind cut, not realized in O_{k}/M_{k}. Observe that there is no a ∈ K such that for every k ∈ ℕ, (Λ − a , Λ′ − a) would induce a Dedekind cut via the residue map in O_{k}/M_{k}. ♢

Finally, we determine the cuts which origin from the upper or lower edge of convex valuation rings or of their maximal ideals:

Theorem 5.18 1) O is a final segment of cΛ + a for some a, c ∈ K, c > 0, if and only if O = O(Λ), the cut (vK \ vI(Λ)) , vI(Λ)) is weakly counter-distinguished, and the cut (Λ, Λ′) is weakly distinguished.

2) O is an initial segment of cΛ′ + a for some a, c ∈ K, c > 0, if and only if O = O(Λ), the cut (vK \vI(Λ)) , vI(Λ)) is weakly counter-distinguished, and the cut (Λ, Λ′) is weakly counter-distinguished.

3) M is a final segment of cΛ + a for some a, c ∈ K, c > 0, if and only if O = O(Λ), the cut (vK \ vI(Λ)) , vI(Λ)) is weakly distinguished, and the cut (Λ, Λ′) is weakly distinguished.

4) M is an initial segment of cΛ′ + a for some a, c ∈ K, c > 0, if and only if O = O(Λ), the cut (vK \ vI(Λ)) , vI(Λ)) is weakly distinguished, and the cut (Λ, Λ′) is weakly counter-distinguished In case 1) and 2), (Λ, Λ′) cannot be projected into O(Λ)/M(Λ). In case 3), we obtain as possible projections all cuts whose initial segment has a last element. In case 4), we obtain as possible projections all cuts whose final segment has a first element.

Proof: We prove part 1); the similar proofs of the other parts are left to the reader. Assume first that O is a final segment of cΛ + a for some a, c ∈ K, c > 0. Then by Corollary 3.6, O = I(cΛ + a). Thus by Lemma 5.13, O(Λ) = O(cΛ + a) = O(I(cΛ + a)) = O(O) = O. As O = I(cΛ + a) is a final segment of cΛ + a, we know from Lemma 3.13 that the cut (cΛ + a , cΛ′ + a) is distinguished. Hence by Lemma 5.2, the cut (Λ, Λ′) is weakly distinguished. Using Lemma 5.1, we compute: cI(Λ) = I(cΛ + a) = O = O(Λ). By Lemma 4.13 it follows that (vK \ vI(Λ)) , vI(Λ)) is weakly counter-distinguished.

To prove the converse, assume that O = O(Λ), that the cut (vK \ vI(Λ)) , vI(Λ)) is weakly counterdistinguished, and that the cut (Λ, Λ′) is weakly distinguished. Then by Lemma 4.13, there is some nonzero c ∈ K such that cI(Λ) = O(I(Λ)) = O(Λ). Since I(Λ) = −I(Λ), we can choose c positive. Since (Λ, Λ′) is weakly distinguished, the same holds for (cΛ, cΛ′) by Lemma 5.2. Now Corollary 3.14 shows that there is some a ∈ K such that I(cΛ) is a final segment of cΛ + a. But by Lemma 5.1, by what we have already proved, and by our assumption on O, we have that I(cΛ) = cI(Λ) = O(Λ) = O. This proves that O is a final segment of cΛ + a for some a, c ∈ K, c > 0. □

Remark 5.19 Let us consider the four cases of the foregoing theorem, for a fixed O. Then we find that the four cuts cannot transformed into each other through transformations of the form (Λ, Λ′) ↦ (cΛ + a , cΛ′ + a), with c > 0. Nevertheless, each of the four cuts can be transformed into the other. We show this for two pairs, the others are similar or combinations of these transformations. 1)↦2): If (Λ, Λ′) is a cut satisfying 1), then −(Λ, Λ′) := (−Λ′, −Λ) is a cut satisfying 2). Indeed, if O is a final segment of cΛ + a, then it is an initial segment of −(cΛ + a) = c(−Λ) − a. 1)↦3): Assume that (Λ, Λ′) is a cut satisfying 1). Then Λ contains positive elements. For this case, we define a cut (Λ, Λ′)^{−1} = (Λ_{1}, Λ′_{1}) as follows. We set Λ_{1} := {d ∈ K | d ≤ 0} ∪ {b^{−1} | b ∈ Λ′} and Λ′_{1} := {b^{−1} | 0 < b ∈ Λ}. Since b ∈ M if and only if b^{−1} ∉ O, it follows that {d ∈ M | d > 0} is a final segment of {b^{−1} | b ∈ Λ′}. Therefore, M is a final segment of Λ_{1} .

The connection between the four cuts can also be understood as follows. Suppose that an element x in some extension field realizes the cut of 1). Then −x realizes the cut of 2), x^{−1} realizes the cut of 3), and −x^{−1} realizes the cut of 4). If we extend the natural valuation to that extension field, we will find that the value γ of x and −x is not archimedean comparable to any value in vK. It lies between H(O) and all values of vK which are smaller than H(O); so γ itself is negative. Further, −γ is the value of x^{−1} and −x^{−1}. It lies between H(O) and all values of vK which are larger than H(O); it is positive.

Exercise. Take any cut (Λ, Λ′) such that Λ contains positive elements. Compute the invariance group and the invariance valuation ring of (Λ, Λ′)^{−1}.

References

[K1] Kuhlmann, F.-V.: Henselian function fields, Ph.D. thesis, Heidelberg (1989)

[K2] Kuhlmann, F.-V. : Valuation theory of fields, abelian groups and modules, to appear in the “Algebra, Logic and Applications” series, eds. A. MacIntyre and R. Göbel, Gordon & Breach, New York [K3] Kuhlmann, F.-V. : Invariance valuation ring of cuts in ordered fields, manuscript, Baton Rouge, (1996)

[KS] Kuhlmann, S. : On the structure of nonarchimedean exponential fields I, Archive for Math. Logic 34 (1995), 145–182

[M] Madden, J. : Two methods in the study of k-vector lattices, dissertation, Wesleyan University (1993)

[R] Ribenboim, P. : Théorie des valuations, Les Presses de l’Université de Montréal (1964)

[T] Tressl, M. : Dedekind cuts in polynomially bounded, o-minimal expansions of real closed fields, doctoral thesis, Regensburg (1996)

[W] Wehrung, F. : Monoids of intervals of ordered abelian groups, preprint, Caen (1994)

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