Places of Near-Fields: To Heinrich Wefelscheid

Abstract

Wefelscheid (Untersuchungen über Fastkörper und Fastbereiche, Habilitationsschrift, Hamburg, 1971) generalised the well-known Theorem of Artin/Schreier about the characterization of formally real fields and the fundamental result of Baer/Krull to near-fields. In the last fifty years arose from the Theorem of Baer/Krull a theory, which analyses the entirety of the orderings of a field (E. Becker, L. Bröcker, M. Marshall et al.), as presented e.g. in the book by Lam (Orderings, valuations and quadratic forms, American Mathematical Society, Providence, 1983). At the centre of this theory are preorders and their compatibility with valuations or places. We develop some essential results of this theory for the near-field case. In particular, we derive the Brown/Marshall’s inequalities and Bröcker’s Theorem on the trivialisation of fans in the near-field case.

1 Notations

By a left-ordering of a near-field F we understand a subset P of $F^*$¹ with the properties

(O1) $P \cup (-P) = F^*$, (O2) $P + P \subset P$ and (O3) $P \cdot P \subset P$.

The associated order relation $< \, = \, <_P$ $(x < y \, \Leftrightarrow \, y - x \in P)$ is then linear and satisfies: From $x< y$ it follows

$$\begin{aligned} a + x< a + y\hbox { for every }a \in F\hbox { and }a \,x < a \,y\hbox { for every }a \in P. \end{aligned}$$

If $x < y$ and $a \in P$ also implies $x \, a < y \, a$, we call P an ordering.

Every left-ordering P of a near-field F contains the square set $F^{(2)} := \{x^2 \,\vert \,x \in F^*\}$. Every intersection T of left-orderings of F satisfies the conditions:

(PO1) $0 \not \in T$, (PO2) $T + T \subset T$, (PO3) $T \, T \subset T$, (PO4) $F^{(2)} \subset T$.

A subset T of the near-field F with (PO1)–(PO4) is called an preorder of F.

A preorder S of a near-field F is called a fan if every subgroup P of index 2 of $F^*$ containing S with $-1 \not \in P$ is a left-ordering of F.

A near-ring $A \subseteq F$ is called a valuation near-ring of F, if $F^* = A^* \cup (A^*)^{-1}$. In this case $M_A := \{x \in F \, \vert \, x^{-1} \not \in A \} \, \cup \, \{0\}$ is an ideal of A, which is the complement of the group of units $U_A$ in A.

The set $\Gamma _A := \{x \, A \,\vert \,x \in F\}$ is linearly ordered by $< \, := \, \subsetneqq$ with smallest element $0 := 0 \, A$. And $v = v_A : x \rightarrow x \, A$ is a valuation on F, i.e.:

• (B1) $v(x) = 0 \, \Leftrightarrow \, x = 0$,

• (B2) $v(x+y) \le \max \{ v(x), \, v(y)\}$    for all $x, \, y \in F$,

• (B3) $v(x) \le v(y) \, \Rightarrow \, v(a \, x) \le v(a \, y)$       for every $a \in F$.

If the ideal $M_A$ has the property

$$\begin{aligned} x, \, y \in A, \, m \in M_A \Rightarrow (x + m) \, y - x \, y \in M_A \, , \end{aligned}$$

the valuation near-ring A is called strict. In the case of a strict valuation near-ring by

$$\begin{aligned} (x+M_A) \cdot (y+M_A) := xy + M_A \ (x, y \in A) \end{aligned}$$

a (well-defined) multiplication is given and $\overline{F}_A := A/M_A$ is then a near-field, the residue near-field of A.

Before we come to the connection between places, valuations and preorders, we introduce in a preparatory Section the Harrisson topology and some invariants for preorders that are essential for everything else, namely the degree of stability and the chain length—all this for the case of a given near-field.

From now on we consider a left-real near-field F.

2 Preparations

2.1 Harrison Topology

For a preorder T of the near-field F, let $X_F/T$ denote the set of all left-orderings P with $P \supseteq T$.

For each $P \in X_F/T$ we denote by $\sigma _T(P)$ the (well-defined) multiplicative character $x \, T \rightarrow {\text {sgn}}_P x$ of $G_T := G_F(T) = F^* / T$.

The mapping $\sigma _T$ is an injection of $X_F/T$ into the character group $\hat{G}_T$ of $G_T$, which can be interpreted as a subset of the Cartesian product $\Pi := \{1, \, -1\}^{G_T}$. Let $\mathfrak {T} = \mathfrak {T}_0^{G_T}$ be the product of the discrete topology $\mathfrak {T}_0$ of $\{1, \, -1\}$ on $\Pi$. The topology $\mathfrak {T}$ is Hausdorffian and, according to the Theorem of Tychonoff compact. A subbasis of $\mathfrak {T}$ is $\{R_1(a) \,\vert \,a \in F^*\} \, \cup \, \{R_{-1} (a) \,\vert \,a \in F^*\}$, where

$$\begin{aligned} R_i(a) := \{ f : G_T \rightarrow \{1, \, -1\} \,\vert \,f(a \, T) = i \, \} \, . \end{aligned}$$

Since

$$\begin{aligned} \sigma _T (P) \in R_1(a) \Leftrightarrow a \in P \Leftrightarrow -a \not \in P \Leftrightarrow \sigma _T(P) \in R_{-1} (-a) \end{aligned}$$

the sets

$$\begin{aligned} H'(a) := R_1(a) \, \cup \, \sigma _T(X_F/T) = \{\sigma _T(P) \,\vert \,a \in P \in X_F/T \, \} \end{aligned}$$

form a subbasis of the trace topology of $\mathfrak {T}$ in $\sigma _T (X_F/T)$. Consequently, the system $\mathfrak {H}_T(F)$ of the Harrison sets

$$\begin{aligned} H_T(a) := \{ P \in X_F/T \,\vert \,a \in P \, \} \end{aligned}$$

forms a subbasis of the initial topology $\mathfrak {T}_H(T)$ of $\sigma _T$ on $X_F/T$. Because of $H_T(-a) = X_F/T {\setminus } H_T(a)$ its elements $H_T(a)$ are open and closed. We call $\mathfrak {T}_H (T)$ the Harrison topology on $X_F/T$. It is obviously the trace topology of $\mathfrak {T}_H(F) := \mathfrak {T}_H (T(F))$ (and the coarsest topology on $X_F/T$ for which all mappings $\varepsilon _a : P \rightarrow {\text {sgn}}_P a$ are continuous).

Lemma 1

$\sigma _T (X_F/T)$ is closed in $(\Pi , \mathfrak {T})$.

Proof

Let $\chi \in \Pi {\setminus } \sigma _T(X_F / T)$. Then we have one of the following cases:

• (1) $\chi (T) = -1$.

• (2) $\chi (a \, T) = \chi (-a \, T)$ for an $a \in F^*$.

• (3) $\chi (a \, T) = 1 = \chi (b \, T)$, and $\chi (a \, b \, T) = -1$ for certain $a, \, b \in F^*$.

• (4) $\chi (a \, T) = 1 = \chi (b \, T)$, and $\chi ((a + b) \, T) = -1$ for certain $a, \, b \in F^*$ with $a+b \in F^*$.

(Otherwise $P := \{x \in F^* \,\vert \,\chi (x \, T) = 1\}$ would be an element of $X_F/T$, and $\chi = \sigma _T(P)$.)

Then there exists a $\mathfrak {T}$-neighbourhood $U \subset \Pi {\setminus } \sigma _T(X_F/T)$ which contains $\chi$:

$$\begin{aligned} U&:= R_{-1}(1) \text { in case } (1) \, , \\ U&:= R_1(a) \cap R_1(-a) \text { respectively } U := R_{-1} (a) \cap R_{-1}(-a) \text { in case } (2) \, , \\ U&:= R_1(a) \cap R_1(b) \cap R_{-1} (a \, b) \text { in case } (3) \text { and } \\ U&:= R_{1} (a) \cap R_{1}(b) \cap R_{-1}(a+b) \text { in case } (4) \, . \end{aligned}$$

$\square$

This gives us the important result:

Theorem 1

For each preorder T of a near-field F, $(X_F/T, \mathfrak {T}_H (T))$ is a Boolean space (i.e. $(X_F/T, \mathfrak {T}_H (T))$ is Hausdorffian, completely disconnected and compact) with subbasis $\mathfrak {H}_T(F) =\{H_T(a) \,\vert \,a \in F^*\}$.

Proof

For elements $P \not = P'$ from $X_F/T$ there exists an $a \in P {\setminus } P'$ such that $P \in H_T(a)$ and $P' \in H_T(-a)$. Because of $H_T(a) \cap H_T(-a) = \emptyset$ and $H_T(a) \cup H_T(-a) = X_F/T$ the topology $\mathfrak {T}_H(T)$ is therefore completely disconnected (and Hausdorffian). And according to Lemma 1$\mathfrak {T}_H(T)$ is compact. $\square$

2.2 Degree of Stability and Chain Length

In this Section, two invariants for preorders are introduced and some of their properties are discussed.

Let a preorder T of a near-field F be given. The factor group $G_T = F^* / T$ is a vector space over $F_2$. Its dimension is denoted by $\delta (T)$. We further denote by

• (A) $\text {st}(T) := \sup \{\delta (S) - 1 \,\vert \,S \in X_F/T \text { is a fan\,}\}$

the degree of stability of T.

Lemma 2

For each preorder T of a near-field F we have:

$$\begin{aligned} {\text {st}}(T) = 0 \ \Leftrightarrow \ T \,\mathrm{is\, a\, left\text {-}ordering}. \end{aligned}$$

For a preorder T and each element $a \in F^*$, $T[a] = T \cup a \, T \cup (T + a \, T)$ is the smallest multiplicatively closed subgroup of $(F,+)$ containing T and a, therefore T[a] is a preorder exactly if $0 \not \in T[a]$; and that is equivalent to $a \not \in -T$. The mapping

$$\begin{aligned} \varepsilon _T: \left\{ \begin{array}{ccc} G_T &{} \rightarrow &{} \mathfrak {H}_T(F)\\ a \, T &{} \rightarrow &{} H_T(a) \end{array} \right. \end{aligned}$$

is well-defined and surjective. From $H_T(a) = H_T(b)$ it follows $a^{-1} \, b \in P$ for every $P \in X_F/T$ and hence $a^{-1} \, b \in T$ according to the Theorem of Artin:

Lemma 3

The mapping $\varepsilon _T: a \, T \rightarrow H_T(a)$ is a bijection of $G_T$ onto $\mathfrak {H}_T(F)$.

The mapping $H_T(a) \rightarrow T[a]$ is well-defined according to the Theorem of Artin, injective and inclusion-reversing. Consequently

• (B) $a \, T < b \, T \ \Leftrightarrow \ T[a] \subsetneqq T[b] \Leftrightarrow H_T(a) \supsetneqq H_T(b)$

defines a (partial) order relation < on $G_T$ with smallest element $T[1] = T$ and largest element $T[-1] = F$.

Following Marshall, we call the element

• (C) $\text {cl}(T) := \sup \{k \in {\mathbb N}\,\vert \,a_0 \, T< a_1 \, T< \cdots < a_k \, T , \, a_i \in F^*, \, k \in {\mathbb N}\}$

f ${\mathbb N}\cup \{\infty \}$ the chain length of T. We note some simple facts:

Lemma 4

For a preorder T of a near-field F we have:

• (a) ${\text {cl}}(T) = 1 \Leftrightarrow T$ is a left-ordering.

• (b) ${\text {cl}}(T) \le 2 \Leftrightarrow T$ is a fan.

• (c) ${\text {cl}}(T) \le \delta (T)$.

Proof

(a) Only $\Rightarrow$ is to be verified. If T is not a left ordering, there exist $P \in X_F/T$ and $a \in P {\setminus } T$. Therefore, we have $T< a \, T < -T$.

(b) Let T be a fan and $a, \, b \in F^*$ with $T< a \, T< b \, T < -T$ be given. It follows $a \in T[a] \subset T[b] \subset T \cup b \, T$, and $a \not \in T$. This shows $a \in b \, T$, i.e. $a \, T = b \, T$. Consequently, we have cl$(T) \le 2$.

If, on the other hand, T is not a fan, then there is an $a \in F^* {\setminus } (-T)$ with $T + a \, T \not \subset T \cup a \, T$. Obviously, we can assume $1 + a \not \in T \cup a \, T$. Then $T< (1 + a) \, T< a \, T < -T$ so that cl$(T) \ge 3$.

(c) Any chain $a_0 \, T< a_1 \, T< \cdots < a_n \, T$ leads to $T[a_0] \subsetneqq T[a_1] \subsetneqq \cdots \subsetneqq T[a_n]$ and thus to a chain $T[a_0]/T \subsetneqq T[a_1]/T \subsetneqq \cdots T[a_n]/T$ of $F_2$-subspaces of $F^*/T$. $\square$

We will also need the following property later [cf. for example [9] (8.13)]:

Theorem 2

[10, (1.7)] Let $T \subset T_i$ ($i = 1, \, \ldots , \, n$) be preorders of F with $X_F/T \subset \bigcup \nolimits _{i = 1}^n X_F/{T_i}$. Then ${\text {cl}}(T) \le \sum \nolimits _{i = 1}^n {\text {cl}}(T_i)$.

Proof

Let $<_i$ denote the order relation on $F^*/T_i$ explained according to (B). From $a \, T \le b\, T$, i.e. $H_T(b) \subset H_T(a)$, it follows $H_{T_i}(b) \subset H_{T_i}(a)$ thus $a \, T_i \le _i b \, T_i$ for $i = 1, \, \ldots , \, n$. Furthermore:

$(*)$ In the case $a \, T < b \, T$ we have $a \, T_i <_i b \, T_i$ for at least one $i \in \{1, \, \ldots , \, n\}$.

Namely, let $a \, T_i = b \, T_i$, i.e. $T_i[a] = T_i[b]$ for $i = 1, \, \ldots , \, n$ be assumed. For each $c \in F^*$, we conclude $X_F/T[c] = \bigcup \nolimits _{i = 1}^n X_F/T_i[c]$ from the premise, so that $T[c] = \bigcap \nolimits _{i = 1}^n T_i[c]$ according to Artin’s Theorem. It follows $T[a] = \bigcap \nolimits _{i=1}^n T_i[a] = \bigcap \nolimits _{i=1}^n T_i[b] = T[b]$ in contrary to the assumption.

Now let a chain $a_0 \, T< a_1 \, T< \cdots < a_k \, T$ be given in $F^*/T$. This induces chains $a_0 \, T_i \le _i a_1 \, T_i \le _i \cdots \le _i a_k \, T_i$ in $F^*/T_i$ for $i = 1, \, \ldots , \, n$. If in the i-th of these chains $k_i$ strict inequalities $<_i$ occur, then with $(*)$ it follows obviously $k \le k_1 + \cdots + k_n \le \text {cl}(T_1) + \cdots + \text {cl}(T_n)$. This establishes $\text {cl}(T) \le \text {cl}(T_1) + \cdots + \text {cl}(T_n)$. $\square$

3 The (Real) Place of a Left-Ordering

For each valuation near-ring A of F, let $\lambda _A$ denote the extension

$$\begin{aligned} x \rightarrow \left\{ \begin{array}{ll} x + M_A &{}\quad \text { if } x \in A \\ \infty &{}\quad \text { if } x \in F {\setminus } A \end{array} \right. \end{aligned}$$

of the projection $\pi _A: A \rightarrow A / M_A$ onto F. We call $\xi := \lambda _A$ the place belonging to A. It obviously has the following properties²:

• (S1) $\xi (x) = \infty \Leftrightarrow x \not = 0 \text { and } \xi (x^{-1}) = 0$.

• (S2) $\xi (x), \, \xi (y) \not = \infty \Rightarrow \xi (x \pm y) = \xi (x) \pm \xi (y)$.

• (S3) $\xi (x), \, \xi (y) \not = \infty \Rightarrow \xi (x \, y) \not = \infty$.

If A is strict, the following stronger condition (S3’) holds:

• (S3’) $\xi (x), \, \xi (y) \not = \infty \Rightarrow \xi (x \, y) = \xi (x) \, \xi (y)$.

On the other hand, if $F' = (F', +)$ is an abelian group, then a surjective mapping $\xi : F \rightarrow F' \cup \{\infty \}$ with the properties (S1), (S2), (S3) is a $F'$-place of F. If $F' = (F', + , \cdot )$ even is a near-field and if (S3’) is satisfied, then we call $\xi$ multiplicative $F'$-place. To $\xi$ it belongs a valuation near-ring, from which $\xi$ essentially arises in this manner:

Lemma 5

Let $F' = (F', +)$ be an abelian group and $\xi$ a $F'$-place of F. Then we have:

• (a) $A_\xi := \xi ^{-1}(F')$ is a valuation near-ring of F with maximal ideal $M_\xi := \xi ^{-1} (\{0\})$.

• (b) The mapping $\varepsilon _\xi : x + M_\xi \rightarrow \xi (x)$ is a group isomorphism of $\overline{F}_\xi := A_\xi /M_\xi$ onto $F'$.

• (c) The valuation near-ring $A_\xi$ is strict if and only if

  $$\begin{aligned} {\text {(i)}} \quad \xi (x) \cdot \xi (y) := \xi (x \, y) \quad (x, \, y \in A_\xi ) \end{aligned}$$

defines a well-defined operation $\cdot$ in $F'$.

Then $(F', +, \cdot )$ is a near-field and $\varepsilon _\xi$ is a near-field isomorphism. (And $\xi$ is a multiplicative $F'$-place.)

The proof is done as in the case of a field.

In analogy to the notations introduced for valuations we call $A_\xi$ valuation near-ring, $M_\xi$ the (maximal) ideal, $U_\xi := A_\xi {\setminus } M_\xi$ the group of units, $\overline{F}_\xi$ the residual class group (in case of strict $A_\xi$ the residual class near-field), $v_\xi := v_{A_\xi } : x \rightarrow x \, A_\xi$ the canonical valuation of the place $\xi$, and we denote $U_\xi ^{(1)} := 1 + M_\xi$. The place $\xi$ is called trivial if $A_\xi$ is trivial. Trivial places yield near-field isomorphisms.

Remark 1

In many cases (e.g. [1, 5, 11]) only multiplicative places $\xi$ with the additional property

$$\begin{aligned} \xi (a \, x - b \, x) \not = \infty ,\quad \xi (x) = \infty \ \Rightarrow \ \xi (a) = \xi (b) \end{aligned}$$

are considered. These are those places whose valuation near-rings Kalhoff [7] denotes by place-near-rings.

The place $\xi$ is called compatible with the left ordering < or P, if $v_\xi$ is compatible with < or P respectively. Due to Karpfinger [8] (2.1)(8) and (5)(b) we have:

Lemma 6

An $F'$-place $\xi$ of F is compatible with a left-ordering < (or P) if and only if by

$$\begin{aligned} x \le y \Leftrightarrow \xi (x) \le _\xi \xi (y) \quad (\mathrm{or }\, P_\xi := \xi (P \cap U_\xi )) \end{aligned}$$

a left order $<_\xi$ (or $P_\xi$) of $(F', +)$ is given.

If $\xi$ is multiplicative, then $<_\xi$ (or $P_\xi$) is a near-field left-ordering. It is an ordering if < is an ordering.

We call $<_\xi$ (resp. $P_\xi$) the left-ordering induced by < and $\xi$ (resp. P and $\xi$).

A place $\xi$ is called compatible or fully compatible with a preorder T in F, if $\xi$ is compatible with at least one or with every left-ordering of $X_F/T$.

It is herewith possible to make a particularly favourable formulation of Karpfinger [8] (2.5), which is going back to Dubois [4] and Brown [3]:

Theorem 3

For every non-archimedean left-ordering < of F there is exactly one isotonic (i.e. compatible with <) and multiplicative ${\mathbb R}$-place³$\lambda _<$ of F, namely

$$\begin{aligned} {\text {(ii)}} \quad \lambda _<: x \rightarrow \left\{ \begin{array}{ll} \sup \{r \in {\mathbb Q}\,\vert \,r< x\} \, , &{} \quad \mathrm{if }\, x \in A_<\\ \infty \, , &{} \quad \mathrm{if } \, x \in F {\setminus } A_<\end{array} \right. . \end{aligned}$$

We have $v_{\lambda _<} = v_<$ and $A_{\lambda _<} = A_<$.⁴

Proof

According to Karpfinger [8] (2.5) $v := v_<$ is strict and non-trivial with the archimedean left-ordered residue class near-field $(\overline{F}_v, <')$.⁵ As every archimedian left-ordered near-field can be embedded into $\mathbb {R}$, there exists an isotonic near-field monomorphism $\varepsilon _<$ from $(\overline{F}_v, <')$ into $({\mathbb R}, <)$. Consequently

$$\begin{aligned} \lambda _< : x \rightarrow \left\{ \begin{array}{lll} \varepsilon _< (x + M_v), &{}\quad \mathrm{if }\, x \in A \\ \infty , &{} \quad \mathrm{if }\, x \in F {\setminus } A \end{array} \right. \end{aligned}$$

is a multiplicative and isotonic ${\mathbb R}$-place. For $x \in A_<$ and $r, \, s \in {\mathbb Q}$ we have

$$\begin{aligned} r< x< s \Rightarrow r = \lambda _< (r) \le \lambda _< (x) \le \lambda _<(s) = s . \end{aligned}$$

This establishes the representation (ii).

If $\xi _1, \, \xi _2$ are two different multiplicative, isotonic ${\mathbb R}$-places of F, then there exists a $x \in F$ with $\xi _1(x) < \xi _2(x)$ and hence an $q \in {\mathbb Q}$ with $\xi _1(x)< q < \xi _2(x)$. Because of $\xi _1(q) = q = \xi _2(q)$ a contradiction arises. $\square$

We call $\lambda _<$ the real place of < and also write $\lambda _{P}$ for $\lambda _<$. The mapping $P \rightarrow \lambda _P$ of X(F) is denoted by $\lambda$. By Karpfinger [8] (3.7), (3.9)(b) and (2.1)(b) it follows:

Theorem 4

For a multiplicative ${\mathbb R}$-place $\xi$ of F the following statements are equivalent:

• (1) $\xi \in \lambda (X_F)$.

• (2) $0 \not \in \sum \xi (Q(F) \cap U_\xi )$.

• (3) $U_\xi ^{(1)} \cap (-T(F)) = \emptyset$.

Remark 2

If F is a field, then each multiplicative ${\mathbb R}$-place of F is in $\lambda (X_F)$. For a skew field F this is in general not true.

Remark 3

The ring $A_{\xi _<} = A_<$ is a place-ring.

Now if $<_1$ and $<_2$ are two non-archimedean left orderings of F with the same image under $\lambda$, then it follows by (3) $A := A_{<_1} = A_{<_2}$ and with the proof of (3) – and the terms there – $\varepsilon _{<_1} \, \pi _A = \lambda _{<_1} = \lambda _{<_2} = \varepsilon _{<_2} \, \pi _A$, such that $\varepsilon _{<_1} = \varepsilon _{<_2}$. And that has $(\overline{F}_{<_1},<_1') = (\overline{F}_{<_2}, <_2')$ as a consequence. This yields:

Theorem 5

For left-orderings $<_1, \, <_2$ of F we have $\lambda _{<_1} = \lambda _{<_2}$ if and only if the natural valuation near-rings $A_{<_1}$ and $A_{<_2}$ coincide and $<_1, \, <_2$ induce the same left-orderings in $\overline{F}_{<_1} = \overline{F}_{<_2}$.

3.1 The Place-Topology

We are now following Lam again [9] §§9, 10, 11. The proofs given by Lam can often be applied literally to the nearfield case. However, because the literature quoted is not so easily accessible and we can occasionally give shorter proofs, we give in this Section all corresponding proofs.

For each preorder T of F we abbreviate $\lambda (X_F/T)$ with $L_F(T)$ and write $L_F$ for $L_F(T(F))$ for short.

The quotient topology of the Harrison topology $\mathfrak {T}_H$ of X(F) with respect to $\lambda$ is denoted by $\mathfrak {T}_L$. Because of Theorem 1 we have:

Lemma 7

$(L_F, \mathfrak {T}_L)$ is compact.

To examine $\mathfrak {T}_L$ in more detail, we introduce for each preorder T of F and each element a of the Prüfer near-ring $A_T = \bigcap \nolimits _{P \in X_F/T} A_P$ [8, Section 3.3] the value function:

$$\begin{aligned} \hbox {(iii)} \quad \varepsilon _a : \left\{ \begin{array}{l} L_F(T) \rightarrow {\mathbb R}\\ \lambda _P \rightarrow \lambda _P(a) \end{array} \right. \end{aligned}$$

Lemma 8

[4] $\varepsilon _a$ is continuous for every preorder T of F and $a \in A_T$.

Here ${\mathbb R}$ is provided with the ordinary topology and $L_F(T)$ with the topology induced by $\mathfrak {T}_L$.

Proof

According to the definition of $\mathfrak {T}_L$ as quotient topology it suffices to show for $a \in A_T$ that $\mu _a := \varepsilon _a \, \lambda : X_F/T \rightarrow {\mathbb R}$ is continuous. This is the case if for each $r \in {\mathbb Q}$ the sets $\mu _a^{-1}(]r, \, +\infty [)$ and $\mu _a^{-1}(]-\infty , \, r[)$ are open in $(X_F/T, \mathfrak {T}_H(T))$. Because of (ii) in Theorem 3, we have for each $P \in X_F/T$:

$$\begin{aligned} P \in \mu _a^{-1}(]r, \, +\infty [)&\Leftrightarrow r< \mu _a(P) = \lambda _P(a) = \sup \{s \in {\mathbb Q}\,\vert \,s<_P a\} \\&\Leftrightarrow r< s<_P a \text { for a } s \in {\mathbb Q}\\&\Leftrightarrow 0< t <_P a - r \text { for a } t \in {\mathbb Q}\, . \end{aligned}$$

Thus $\mu _a^{-1} (]r, \, + \infty [) = \bigcup \nolimits _{0 < t \in {\mathbb Q}} H_T (a - r-t)$ is open in $(X_F/T, \mathfrak {T}_F(T))$.

Similarly, one shows that $\mu _a^{-1}(]-\infty , \, r[)$ is open. $\square$

We also need:

Lemma 9

The set $\{\varepsilon _a \,\vert \,a \in A_{T(F)}\}$ separates different points in $L_F$.

Proof

Let there be given different $\xi , \, \xi ' \in L_F$. Then there exists an $x \in F$ with $\xi (x) < \xi '(x)$.

1. case: $\xi '(x) \not = \infty$. There is an $r \in {\mathbb Q}$ with $\xi (x)< \xi (r) = r = \xi '(r) < \xi '(x)$. For $y := x - r$ it follows $\xi (y)< 0 < \xi '(y)$. According to Karpfinger [8], (3.14), $a := (y + y^{-1})^{-1}$ is in $A_{T(F)}$, and $\xi (a)< 0 < \xi '(a)$.

2. case: $\xi '(x) = \infty$. We have $0 < \xi (x + n)$ for a suitable $n \in {\mathbb N}$. For $y := x + n$ it follows $0< \xi (y) < \xi '(y) = \infty$ and hence $\xi '(y^{-1})< \xi (y^{-1}) < \infty$, so that $\xi (-y^{-1})< \xi '(-y^{-1}) < \infty$. Thus, we have again case 1 (for $x := -y^{-1}$). $\square$

With these results we now get:

Lemma 10

[4]

• (a) The space $(L_F, \mathfrak {T}_L)$ is Hausdorffian.

• (b) The mapping $\lambda$ is continuous and closed.

• (c) For every preorder T of F, $L_F(T)$ is a compact subset of $L_F$.

Proof

(a) According to Lemma 9, for $\xi \not = \xi '$ in $L_F$ there is an $a \in A_{T(F)}$ with $\xi (a) \not = \xi '(a)$. If $I, \, I'$ are disjoint open intervals of ${\mathbb R}$ with $\xi (a) \in I$ and $\xi '(a) \in I'$, then, by Lemma 8, $\varepsilon _a^{-1}(I)$ and $\varepsilon _a^{-1}(I')$ are disjoint environments of $\xi$ and $\xi '$, respectively.

(b) follows from (a), Theorem 1 and the continuity of $\lambda$.

(c) By Theorem 1, $X_F/T$ is a compact subset of $(X_F, \mathfrak {T}_H)$. Because of the continuity of $\lambda$, the assertion follows. $\square$

A concrete description of $\mathfrak {T}_L$ is now possible:

Lemma 11

[4]

• (a) The topology $\mathfrak {T}_L$ is the initial topology of the mappings $\varepsilon _a$ ($a \in A_{T(F)}$).

• (b) The sets $H(a) := \{\xi \in L_F \,\vert \,\xi (a) > 0\}$ with $a \in A_{T(F)}$ form a subbasis of $\mathfrak {T}_L$.

Proof

(a) Let $\mathfrak {T}'$ be the initial topology of the mappings $\varepsilon _a$ ($a \in A_{T(F)}$), and $\mathfrak {T} := \mathfrak {T}_L$. According to Lemma 8, $\mathfrak {T}$ is finer than $\mathfrak {T}'$. Since $\mathfrak {T}$ is compact by Lemma 7, the topology $\mathfrak {T}'$ is compact, too. Since the mappings $\varepsilon _a$ separate points in $L_F$ and are continuous with respect to $(\mathfrak {T}_H, \mathfrak {T}'$), it follows literally as in the proof of Lemma 10(a), that $\mathfrak {T}'$ is also Hausdorffian. Consequently, ${\text {Id}}: (L_F, \mathfrak {T}) \rightarrow (L_F, \mathfrak {T}')$ is continuous and closed, i.e. a homeomorphism: $\mathfrak {T} = \mathfrak {T}'$.

(b) According to Lemma 8 the sets $H(a) = \varepsilon _a^{-1}(]0, \, +\infty [)$ are open with respect to $\mathfrak {T} = \mathfrak {T}_L$. On the other hand, the sets $\varepsilon _a^{-1}(]r, \, s[)$ with $a \in A_{T(F)}$ and $r < s$ in ${\mathbb Q}$ form a subbasis of $\mathfrak {T}' = \mathfrak {T}$ ($\mathfrak {T}'$ as in the proof of (a)). Furthermore,

$$\begin{aligned} \varepsilon _a^{-1}(]r, \, s [)&= \!\!\{ \lambda _P \in \lambda (X(F)) \,\vert \,r< \lambda _P(a) < s\} \\&= \!\!\{\lambda _P \in \lambda (X(F)) \!\,\vert \,\! \lambda _P(a - r)> 0\} \cap \{\lambda _P \in \lambda (X(F)) \!\,\vert \,\! \lambda _P (s - a) > 0\} \\&= \!\!H(a-r) \cap H(s-a) \, , \end{aligned}$$

and $a-r, \, s - a\in A_{T(F)}$. This proves the assertion. $\square$

Remark 4

Unlike $\mathfrak {T}_H$, $\mathfrak {T}_L$ need not be totally disconnected at all. There are even fields F for which $\mathfrak {T}_L$ is connected.

According to Lemma 8, for each preorder T of F the value functions $\varepsilon _a$ ($a \in A_T$) lie in the ring $C_T := C(L_F(T), \, {\mathbb R})$ of all continuous functions of $L_F(T)$ in ${\mathbb R}$. Since $L_F(T)$ is compact according to Lemma 10 (c), the supremum norm

$$\begin{aligned} \hbox {(iv)} \quad \Vert f \Vert _T := \sup \{ \vert f(\lambda _P ) \vert \,\vert \,\lambda _P \in L_F(T) \} \quad (f \in C_T) \end{aligned}$$

is defined on $C_T$.

Lemma 12

For each preorder T of F the set $\varepsilon (A_T) = \{\varepsilon _a \,\vert \,a \in A_T\}$ lies dense in $C_T$ with respect to $\Vert \, \Vert _T$.

Proof

Because of Lemma 9$\varepsilon (A_T)$ separates points in $L_F(T)$; and the constant function $\varepsilon _1: \xi \rightarrow 1$ lies in $\varepsilon (A_T)$. The assertion therefore follows with the Theorem of Stone/Weierstrass. $\square$

This provides:

Theorem 6

(Separation criterion) For each preorder T of F and disjoint closed subsets $A, \, B$ of $X_F/T$ are equivalent:

• (1) $\lambda (A) \cap \lambda (B) = \emptyset$.

• (2) There exists an $x \in A_T \cap \bigcap \nolimits _{\xi \in \lambda (A \cup B)} U_\xi$ with $A \subset H_T(x)$ and $B \subset H_T(-x)$.

• (3) There exists an $x \in A_T \cap \bigcap \nolimits _{\xi \in \lambda (A)} U_\xi$ with $A \subset H_T(x)$ and $B \subset H_T(-x)$.

Proof

(1) $\Rightarrow$ (2): According to Lemma 10$\lambda (A)$ and $\lambda (B)$ are closed. By Urysohn’s lemma and due to $\lambda (A) \, \cap \, \lambda (B) = \emptyset$, there is a continuous function $f: X_F/T \rightarrow {\mathbb R}$ with $f(\xi ) = 1$ for all $\xi \in \lambda (A)$ and $f(\zeta ) = -1$ for all $\zeta \in \lambda (B)$. And according to Lemma 12 there exists an $x \in A_T$ with $\Vert \varepsilon _x -. f\Vert _T < 1$, i.e.

$$\begin{aligned} \hbox {(v)} \quad \vert \lambda _P(x) - f(\lambda _P)\vert < 1 \ \text { for all } \lambda _P \in \lambda (X(F)). \end{aligned}$$

For every $P \in A$ we have $\lambda _P \in \lambda (A)$ so that $\vert \lambda _P(x) - 1 \vert < 1$ due to (v), and it follows $\lambda _P(x) > 0$. This justifies $x \in U_{\lambda _P}$ and $x \in P$ according to Theorem 3. Similarly, for any $P \in B$, i.e. $\lambda _P \in \lambda (B)$, we have $\vert \lambda _P (x) + 1 \vert < 1$ due to (v) and therefore shows $\lambda _P(x) < 0$. This has $x \in U_{\lambda _P}$ and—again according to Theorem 3—$x \in -P$ as a consequence.

(2) $\Rightarrow$ (3) is trivially correct.

(3) $\Rightarrow$ (1): Suppose $\lambda (P) = \lambda (Q)$ for certain $P \in A$ and $Q \in B$. And let x be chosen as in (3). Since $x \in U_{\lambda _P} = U_{\lambda _Q}$ and $x \in P$ and $x \in -Q$ we get a contradiction $\lambda _P(x) > 0$ to Theorem 3 and $\lambda _Q(x) < 0$. $\square$

An interesting consequence is:

Corollary 1

For each two preorders $T, \, T'$ of F we have

$$\begin{aligned} \lambda (X_F/(T \cap T')) = \lambda (X_F/T) \cup \lambda (X_F/T'). \end{aligned}$$

Proof

The inclusion $\lambda (X_F/T) \, \cup \, \lambda (X_F/T') \subset \lambda (X_F/(T \, \cap \,T'))$ is trivially correct. Suppose there is an $P \in X_F/(T \, \cap \, T')$ with $\lambda (P) \not \in \lambda (X_F/T) \, \cup \, \lambda (X_F/T') = \lambda (X_F/T \, \cup \, X_F/T')$. According to Theorem 6 [(1) $\Rightarrow$ (2)], $\{P\}$ can be separated from the set $B := X_F/T \, \cup \, X_F/T'$ which is closed by Theorem 1: There exists an $x \in P$ with $-x \in Q$ for each $Q \in B$. According to Artin’s Theorem it follows $-x \in T \cap T'$ in contradiction to $T \cap T' \subset P$. $\square$

We derive a corollary for natural evaluation near-rings and Prüfer near-rings:

Corollary 2

For each two preorders $T, \, T'$ of F we have

$$\begin{aligned} A_{T \cap T'} = A_T \cap A_{T'} \quad \mathrm{and }\quad \ A^{T \cap T'} =. A^T \, A^{T'}. \end{aligned}$$

⁶

Proof

The inclusions $A_{T \cap T'} \subset A_T \cap A_{T'}$ and $A^T \, A^{T'} \subset A^{T \cap T'}$ are clear. On the other hand, according to Corollary 1 for each $P \in X_F/(T \cap T')$ $A_P$ coincides with $A_{P'}$ for any $P' \in X_F/T \, \cup \, X_F/T$. From this the reverse inclusions follow. $\square$

If T is a preorder of F, then $\{X_\xi \,\vert \,\xi \in L_F(T)\}$ with

$$\begin{aligned} \hbox {(vi)} \quad X_\xi := \{P \in X_F/T \,\vert \,\lambda _P = \xi \} \end{aligned}$$

forms a decomposition of $X_F/T$ according to Theorem 3. We now show that $X_\xi = X_F/T_\xi$ for some fan $T_\xi$ with $T_\xi \supseteq T$.

Let $P \in X_F/T$ be a left-ordering and $\xi := \lambda _P$. It follows from Theorem 3 and the fact that $\xi = \varepsilon _\xi \, \pi _{v_\xi }$ [cf. Lemma 5 (b)] and [8], (3.2) that:

$$\begin{aligned} T_\xi := T \, \xi ^{-1} (\overline{F}^*_\xi \cap {\mathbb R}^{(2)}) = T \, \xi ^{-1} (\xi (U_\xi \cap P)) = T \, \pi _{v_\xi }^{-1}(\pi _{v_\xi }(U_\xi \cap P)) = T \wedge \pi _{v_\xi }(P) . \end{aligned}$$

⁷

And according to Karpfinger [8], (3.2), the preorder $T_\xi$ of F is fully compatible with $\pi _{v_\xi }$ (i.e. with $\xi$) and $\pi _{v_\xi }(T_\xi ) = \pi _{v_\xi }(P)$, i.e. due to Theorem 3: $\xi (T_\xi \cap U_\xi ) = \overline{F}^*_\xi \cap {\mathbb R}^{(2)}$. According to Karpfinger [8], (2.3) (b) $\pi _{v_\xi }(P)$ is a left-ordering of $\overline{F}_\xi$; and by Karpfinger [8], (3.10) $T_\xi$ is a fan of F. This shows the first of the following statements:

Lemma 13

For each preorder T of F and each place $\xi \in L_F(T)$ we have:

• (a) $T_\xi := T \, \xi ^{-1}(\overline{F}_\xi ^* \cap {\mathbb R}^{(2)})$ is a fan of F fully compatible with $\xi$; and $\xi (T_\xi \cap U_\xi ) = \xi (P \cap U_\xi ) = \overline{F}^*_\xi \cap {\mathbb R}^{(2)}$ for each $P \in X_F/T$ with $\xi = \lambda _P$.

• (b) $X_\xi = X_F/T_\xi$, i.e. for $P \in X_F/T$ we have: $T_\xi \subset P \Leftrightarrow \lambda _P = \xi$.

• (c) $X_F/T$ is the disjoint union of the sets $X_F/T_\xi$ with $\xi \in L_F(T)$.

Proof

(b) From $P \in X_F/T$ and $\lambda _P = \xi$ it follows with (a):

$$\begin{aligned} T_\xi = T \, \xi ^{-1} (\overline{F}_\xi ^* \cap {\mathbb R}^{(2)}) = T \, \xi ^{-1} (\xi (P\cap U_\xi )) \subset T \, P = P \end{aligned}$$

(since $\xi (P\cap U_\xi ) \subset {\mathbb R}^{(2)}$).

Thus, $X_\xi \subset X_F/T_\xi$. On the other hand, any left-ordering P of $X_F/T_\xi$ is compatible with $\xi$ by (a). Therefore, due to Theorem 3 we have $\lambda _P = \xi$, i.e. $P \in X_\xi$.

(c) follows directly from (b). $\square$

We now prove the Brown/Marshall’s inequalities (cf. for example Lam [9] 10.10).

Theorem 7

For each preorder T of F we have

$$\begin{aligned} \vert L_F(T) \vert \le {\text {cl}}(T) \le 2 \, \vert L_F(T)\vert . \end{aligned}$$

Proof

To prove the first inequality, let $P_1, \, \ldots , \, P_n \in X_F/T$ with different real places $\lambda _{P_1}, \, \ldots , \, \lambda _{P_n}$ be given. We construct inductively elements $a_0, \, \ldots , \, a_n \in A_T$ with the properties:

• (a) $a_0 = -1$.

• (b) $a_i \in U_{P_1} \cap \ldots \cap U_{P_n}$.

• (c) For each $i = 1, \, \ldots , \, n$ $a_i$ is positive with respect to the left-orderings in $H_T(a_{i-1}) \cup \{P_i\}$ and negative with respect to $P_{i+1}, \, \ldots , \, , P_n$.

It then follows $\emptyset = H_T(a_0) \subsetneqq H_T(a_1) \subsetneqq \cdots \subsetneqq H_T(a_n)$. According to Sect. 2.2, (B) and (C), this results in $n \le {\text {cl}}(T)$, which proves the first inequality.

Now, if $a_0, \, \ldots , \, a_{i-1}$ ($i \ge 0$) with the properties (a)–(c) are already constructed, then it follows with Theorem 6 [(3) $\Rightarrow$ (1)]: $\lambda (H_T(a_{i-1})) \, \cap \lambda (\{P_{i+1}, \, \ldots , \, P_n\}) = \emptyset$, since $a_{i-1}$ separates the disjoint and closed sets $A := \{P_{i+1}, \, \ldots , \, P_n\}$ and $B := H_T(a_{i-1})$ and lies in $U_{P_{i+1}} \cap \ldots \cap U_{P_n}$. Then also $\lambda (A) \cap \lambda (B') = \emptyset$ for $B' := H_T(a_{i-1}) \cup \{P_i\}$. With Theorem 6 [(1) $\Rightarrow$ (2)]—for $B'$ instead of B—it follows that there exists an element $a_i$ with properties (b) and (c).

To prove ${\text {cl}}(T) \le 2 \, \vert L_F(T) \vert$ we may assume $L_F(T)$ as finite. According to Lemma 13$X_F/T$ is a (disjoint) union $X_F/T = X_F/T_{\xi _1} \cup \cdots \cup X_F/T_{\xi _n}$ with places $\xi _i \in L_F(T)$; and each $T_{\xi _i}$ is a (T covering) fan. Due to Theorem 2 and Lemma 4(b) we have

$$\begin{aligned} {\text {cl}}(T) \le 2 \, ({\text {cl}}(T_{\xi _1}) + \cdots + {\text {cl}}(T_{\xi _n})) \le 2 \, n. \end{aligned}$$

$\square$

From Theorem 7 and Lemma 4(b) one obtains directly:

Corollary 3

For each fan S of F we have $\vert L_F(S)\vert \le 2$.

We can now prove Bröcker’s Theorem [2] about the trivialisation of fans (c.f. Kalhoff [6]):

Theorem 8

For each fan S of F for which $X_F/S$ contains at least one ordering, there is a place $\xi : F \rightarrow F' \cup \{\infty \}$ fully compatible with S for which $\xi (S \cap U_\xi )$ is a trivial fan⁸ of $F'$.

Proof

If S is a trivial fan, we choose the trivial place $\xi$. Let S therefore be nontrivial. According to corollary 3, two cases are possible:

1. Case: $L_F(S) = \{\xi \}$ with non-trivial $\xi$. In this case, any $P \in X_F/S$ is compatible with $\xi$ and by Lemma 13 (and the Theorem of Artin) it follows $S = T_\xi$ and $\xi (S \cap U_\xi ) = \overline{F}^*_\xi \cap {\mathbb R}^{(2)}$; and that is a left-order of $\overline{F}_\xi$.

2. Case: $L_F(S) = \{\eta , \, \zeta \}$ with different places $\eta , \, \zeta$, and without loss of generality $\zeta = \lambda _Q$ for some ordering Q of $X_F/S$. All $P \in X_F/S$ are non-archimedean, i.e. $\eta , \, \zeta$ are non-trivial, and topologically equivalent. By Karpfinger [8], (2.9) it follows that $A := A_\eta \, A_\zeta$ is a strict valuation near-ring $\not = F$ compatible with any $P \in X_F/S$. Now $\xi = \lambda _A$ is nontrivial. According to Karpfinger [8], (3.10) and Lemma 5, $S' := \xi (S \cap U_\xi )$ is a $\xi$-fan of the near-field $\overline{F}_\xi$. If $S'$ were non-trivial, then a non-trivial place $\xi ' : \overline{F}_\xi \rightarrow F' \cup \{\infty \}$ fully compatible with $S'$ exists. This is shown in the last part of the proof applied to $\overline{F}_\xi$, since $\xi (Q \cap U_\xi )$ is an ordering of $\overline{F}_\xi$ due to Karpfinger [8], (2.3) (b). Then—with $\xi '(\infty ) := \infty$—$\xi ' \, \xi : F \rightarrow F' \cup \{\infty \}$ is a place of F fully compatible with S. Since $\xi '$ is nontrivial, we have $A_{\xi ' \, \xi } \subsetneqq A_\xi$. Since $\xi ' \, \xi$ is compatible with all $P \in X_F/S$, we have on the other hand $A_\eta , \, A_\zeta \subset A_{\xi '\, \xi }$ and hence we get the contradiction $A_\xi = A = A_\eta \, A_\zeta \subset A_{\xi '\, \xi }$. $\square$