## Definition ## erm... ### With strict order ### An __ordered integral domain__ is a strictly ordered integral $\mathbb{Z}$-algebra such that for every element $a:A$ such that $\max(a, -a) \gt 0$, there is an element $b:A$ such that $a \cdot b = 1$. ### With positivity ### An ordered integral domain is a [[Heyting integral domain]] 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 $$\prod_{a:A} \prod_{b:A} \mathrm{isPositive}(a) \times \mathrm{isPositive}(b) \to \mathrm{isPositive}(a \cdot b)$$ ## Examples ## * The [[integers]] are an ordered integral domain. * The [[rational numbers]] are a ordered integral domain * Every [[ordered field]] is a ordered integral domain. ## See also ## * [[Heyting integral domain]] * [[ordered field]] * [[unit interval]] * [[Archimedean ordered integral domain]]