Ordered fields

# Ordered fields

## Idea

An ordered field is a field equipped with a compatible strict total order.

Note that while the adjective ‘ordered’ usually refers to a partial order, it is traditionally used more strictly when placed before ‘field’.

## Definition

Classically:

An ordered field is a field $K$ equipped with a strict total order $\lt$ such that

• $0 \lt 1$

• for all $a \in R$ and $b \in R$, $0 \lt a$ and $0 \lt b$ implies that $0 \lt a + b$; alternatively, $0 \lt a + b$ implies that $0 \lt a$ or $0 \lt b$.

• for all $a \in R$ and $b \in R$, if $0 \lt a$ and $0 \lt b$, then $0 \lt a \cdot b$

One often sees the definition using a weak total order $\leq$ instead of the strict total order $\lt$. This makes no difference in classical mathematics, but the definition of the strict total order is the one that generalizes to constructive mathematics.

An ordered field could also be defined as a field with a predicate $\mathrm{isPositive}$ such that

• zero is not positive:
$\not \mathrm{isPositive}(0)$
• one is positive:
$\mathrm{isPositive}(1)$
• for every element $a:A$, if $a$ is not positive and $-a$ is not positive, then $a = 0$
$\forall a:A.\neg\mathrm{isPositive}(a) \wedge \neg\mathrm{isPositive}(-a) \Rightarrow (a = 0)$
• for every term $a:A$, if $a$ is positive, then $-a$ is not positive.
$\forall a:A. \forall b:A. \mathrm{isPositive}(a) \Rightarrow \neg\mathrm{isPositive}(-a)$
• for every term $a:A$, $b:A$, if $a$ is positive, then either $b$ is positive or $a - b$ is positive.
$\forall a:A. \forall b:A. \mathrm{isPositive}(a) \Rightarrow \left(\mathrm{isPositive}(b) \vee \mathrm{isPositive}(a - b)\right)$
• for every term $a:A$, $b:A$, if $a$ is positive and $b$ is positive, then $a + b$ is positive
$\forall a:A. \forall b:A. \mathrm{isPositive}(a) \wedge \mathrm{isPositive}(b) \Rightarrow \mathrm{isPositive}(a + b)$
• for every term $a:A$, $b:A$, if $a$ is positive and $b$ is positive, then $a \cdot b$ is positive
$\forall a:A. \forall b:A. \mathrm{isPositive}(a) \wedge \mathrm{isPositive}(b) \implies \mathrm{isPositive}(a \cdot b)$

The strict total order is defined as

$a \lt b \coloneqq \mathrm{isPositive}(b - a)$

### In constructive mathematics

In constructive mathematics, denial inequality is no longer a tight apartness relation, and neither relation, is a decidable relation. Thus, the concept of a field splits into multiple separate notions, discrete fields, Heyting fields, and Kock fields, depending on whether one uses denial inequality or a separate tight apartness relation to characterize invertibility of elements. This split in the definition of field carries over to the definition of an ordered field.

• An ordered discrete field is an ordered local ring such that $x$ is invertible or $x = 0$.

• An ordered Heyting field is an ordered local ring such that $x$ is non-invertible if and only if $x = 0$.

• An ordered Kock field is an ordered local ring such that $x$ is invertible if and only if $\neg (x = 0)$.

Ordered Heyting fields are commonly used in real analysis with the Dedekind real numbers, since the Dedekind real numbers $\mathbb{R}$ and any subfield of $\mathbb{R}$ form an ordered Heyting field. On the other hand, ordered Kock fields are commonly used in synthetic differential geometry.

Furthermore, a set having a strict total order no longer implies that the set has a weak total order, and a set having a weak total order no longer implies that the set has a strict total order. In particular, the traditional definition of an ordered field as defined above no longer implies that the ordered field is a lattice, that it has binary joins and meets.

Thus, some authors in constructive mathematics, such as Booij 2020 and Univalent Foundations Project 2013, have defined an ordered field to additionally have a lattice structure on $\leq$. However, it is consistent in constructive mathematics that there are ordered Heyting fields without a lattice structure (see here for example), and ordered Kock fields which satisfy the Kock-Lawvere axiom are not lattices with respect to $\leq$ either.

## Examples

The field $\mathbb{R}$ of real numbers is the Dedekind-complete ordered field.

The field $\mathbb{Q}$ of rational numbers is a subfield of $\mathbb{R}$ that is too small to be complete.

The field of surreal numbers is a field extension of $\mathbb{R}$ that is too large to be complete.

## Properties

Every ordered field must have characteristic $0$, since we can prove by induction that $n \gt 0$ for every positive natural number $n$.

As a result, the rational numbers are the initial ordered field, and every ordered field is a $\mathbb{Q}$-algebra.

The archimedean ordered fields are precisely the subfields of the field of real numbers.

###### Proposition

Every Dedekind complete ordered field is archimedean.

###### Proof

Suppose otherwise: let $a, b \gt 0$ be given, and suppose $b$ is an upper bound of $a, 2a, 3a, \ldots$. Then $b - a$ is an upper bound of $0, a, 2a, \ldots$ and consequently there can be no least upper bound of the sequence, contradicting Dedekind completeness.

The following is a result in classical mathematics.

###### Proposition

A field admits an order (“is orderable”) if and only if it is a real field, i.e., if the element $-1$ is not a sum of squares.

###### Proof

Given an ordered field, any non-zero square is positive since either $-\alpha$ or $\alpha$ is positive, and so $(-\alpha)^2 = \alpha^2$ is positive. Hence a sum of non-zero squares cannot be negative, and in particular cannot be equal to $-1$.

In the other direction, every real field $F$ may be embedded in a real closed field (this requires Zorn's lemma), and a real closed field admits a unique ordering. The restriction of this ordering to the embedded field $F$ gives an ordering on $F$.