nLab Archimedean ordered integral domain

Idea

An Archimedean ordered integral domain is an ordered integral domain that satisfies the Archimedean property.

Properties

Every Archimedean ordered integral domain extension of the integers $\mathbb{Z}[x]$ is a dense linear order.

Since $\mathbb{Z}[x]$ is Archimedean, it does not have either infinite or infinitesimal elements, which means there exists an integer $a$ and natural number $b$ such that $0 \lt x - a \lt 1$ and $1 \lt b(x - a)$. $0 \lt x - a \lt 1$ implies $0 \lt (d - c)(x - a) \lt d - c$ and $c \lt (d - c)(x - a) + c \lt d$ for all $c$ and $d$ in $\mathbb{Z}[x]$. Let $y = (d - c)(x - a) + c$. Since there exists an element $y$ such that $c \lt y \lt d$ for all $c$ and $d$, $\mathbb{Z}[x]$ is a dense linear order.

This means that the Dedekind completion of every Archimedean ordered integral domain extension of the integers is the integral domain of real numbers.

Examples

Archimedean ordered integral domains include

• integers
• rational numbers
• decimal rational numbers
• dyadic rational numbers
• real numbers

Non-Archimedean ordered integral domains include

• p-adic integers.

Related concepts

• Archimedean ordered field

• Archimedean group

• archimedean difference protoring

• enriched set