# Homotopy Type Theory ordered integral domain > history (changes)

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## Definition

An

ordered integral domain is a totally ordered commutative ring? which comes with a strict order $\lt$ such that

• $0 \lt 1$
• for all elements $a:R$ and $b:R$, if $0 \lt a$ and $0 \lt b$, then $0 \lt a + b$
• for all elements $a:R$ and $b:R$, if $0 \lt a$ and $0 \lt b$, then $0 \lt a \cdot b$
• for all elements $a:R$ and $b:R$, if $0 \lt \max(a, -a)$ and $0 \lt \max(b, -b)$, then $0 \lt \max(a \cdot b, -a \cdot b)$

### With positivity

An ordered field is a commutative ring with a predicate $\mathrm{isPositive}$ such that

• zero is not positive:
$\mathrm{isPositive}(0) \to \emptyset$
• one is positive:
$\mathrm{isPositive}(1)$
• for every term $a:A$, if $a$ is not positive and $-a$ is not positive, then $a = 0$
$\prod_{a:A} ((\mathrm{isPositive}(a) \to \emptyset) \times (\mathrm{isPositive}(-a) \to \emptyset)) \to (a = 0)$
• for every term $a:A$, if $a$ is positive, then $-a$ is not positive.
$\prod_{a:A} \prod_{b:A} \mathrm{isPositive}(a) \to (\mathrm{isPositive}(-a) \to \emptyset)$
• for every term $a:A$, $b:A$, if $a$ is positive, then either $b$ is positive or $a - b$ is positive.
$\prod_{a:A} \prod_{b:A} \mathrm{isPositive}(a) \to \left[\mathrm{isPositive}(b) + \mathrm{isPositive}(a - b)\right]$
• for every term $a:A$, $b:A$, if $a$ is positive and $b$ is positive, then $a + b$ is positive
$\prod_{a:A} \prod_{b:A} \mathrm{isPositive}(a) \times \mathrm{isPositive}(b) \to \mathrm{isPositive}(a + b)$
• for every term $a:A$, $b:A$, if $a$ is positive and $b$ is positive, then $a \cdot b$ is positive
$\prod_{a:A} \prod_{b:A} \mathrm{isPositive}(a) \times \mathrm{isPositive}(b) \to \mathrm{isPositive}(a \cdot b)$
• for every term $a:A$, if $a$ is positive, then there exists a $b$ such that $a \cdot b = 1$ and $b \cdot a = 1$
$\prod_{a:A} \mathrm{isPositive}(a) \to \left[\sum_{b:A} (a \cdot b = 1) \times (b \cdot a = 1)\right]$