# Homotopy Type Theory ordered abelian group > history (changes)

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## Definition

### With strict order

An ordered abelian group is an abelian group $A$ with a strict order $\lt$ and a family of dependent terms

$a:A, b:A \vdash \alpha(a, b): (0 \lt a) \times (0 \lt b) \to (0 \lt a + b)$

### With positivity

An ordered abelian group is an abelian group $A$ with a predicate $\mathrm{isPositive}$ such that

• zero is not positive:
$\mathrm{isPositive}(0) \to \emptyset$
• 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
$\prod_{a:A} \prod_{b:A} \mathrm{isPositive}(a) \times \mathrm{isPositive}(b) \to \mathrm{isPositive}(a + b)$