# Contents

## Idea

In the same way that a local ring is a Heyting field whose apartness relation is not tight, an ordered local ring is an ordered field whose strict order is not necessarily linear.

## Definition

### With an order relation

Let $R$ be a commutative ring. $R$ is an ordered local ring if there is a binary relation $\lt$ such that

• for all $a \in R$, $\neg(a \lt a)$

• for all $a \in R$ and $b \in R$, if $a \lt b$, then $\neg(b \lt a)$

• for all $a \in R$, $b \in R$, and $c \in R$, if $a \lt c$, then $a \lt b$ or $b \lt c$

• $0 \lt 1$

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

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

• for all $a \in R$, $a$ is invertible if and only if $a \lt 0$ or $0 \lt a$

We define the predicate $\mathrm{isPositive}$ as

$\mathrm{isPositive}(a) \coloneqq 0 \lt a$

### With a positivity predicate

Let $R$ be a commutative ring. $R$ is an ordered local ring if there is a predicate $\mathrm{isPositive}$ stating an element $a$ is positive, such that

• 0 is not positive
• given element $a \in R$, if $a$ is positive, then $-a$ is not positive
• given elements $a \in R$ and $b \in R$; if $a$ is positive, then either $b$ is positive or $a - b$ is positive
• 1 is positive
• given elements $a \in R$ and $b \in R$, if $a$ is positive and $b$ is positive, $a + b$ is positive
• given elements $a \in R$ and $b \in R$, if $a$ is positive and $b$ is positive, $a \cdot b$ is positive
• for all $a \in R$, $a$ is invertible if and only if $a$ is positive or $-a$ is positive

We define the order relation $\lt$ as

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

## Properties

Every ordered local ring is a local ring with the apartness relation given by

$a \# b \coloneqq a \lt b \vee b \lt a$

Every ordered local ring has a preorder given by $a \leq b \coloneqq \neg (b \lt a)$.

### Quotient ordered field

Let $D$ be the ideal of all non-invertible elements in $R$. Then the quotient ring $R/D$ is an ordered field.

## Examples

• Every ordered field is an ordered local ring where every non-positive non-negative element is equal to zero.

• Every ordered Kock field is an ordered local ring.

• The dual numbers $\mathbb{R}[\epsilon]/\epsilon^2$ are an ordered local ring where the nilpotent infinitesimal $\epsilon \in \mathbb{R}[\epsilon]/\epsilon^2$ is a non-zero non-positive non-negative element.