Contents

group theory

# Contents

## Idea

An archimedean group is a linearly ordered group in which every positive element is bounded above by a natural number.

So an archimedean group has no infinite elements (and thus no non-zero infinitesimal elements).

## Definition

Let $(\mathbb{N}^+,1:\mathbb{N}^+,s:\mathbb{N}^+\to \mathbb{N}^+)$ be the set of positive integers.

Let $(A,\lt)$ be a linearly ordered group. The positive integers are embedded into the function group $A \to A$; there is an injection $inj:\mathbb{N}^+\to (A \to A)$ such that $inj(1) = id_A$ and $inj(s(n)) = inj(n) + id_A$ for all $n:\mathbb{N}^+$.

The archimedean property states that for every $a,b:A$ such that $0 \lt a$ and $0 \lt b$, then there exist $n:\mathbb{N}^+$ such that $a \lt inj(n)(b)$.

By uncurrying $inj$ one gets an action $act: (\mathbb{N}^+\times A) \to A$ such that $act(1,a) = a$ and $act(s(n),a) = act(n,a) + a$ for all $n:\mathbb{N}^+$ and $a:A$. The archimedean property then states that for all $a,b:A$ such that $0 \lt a$ and $0 \lt b$, there exist $n:\mathbb{N}^+$ such that $a \lt act(n,b)$.

An archimedean group is a linearly ordered group satisfying the archimedean property.

An archimedean group that is also a linearly ordered ring is called an archimedean ring. If an archimedean ring is a $R$-algebra, then it is called an archimedean $R$-algebra, and if it is a field, then it is called an archimedean field.

## Properties

Every archimedean group is an abelian group and has no bounded cyclic subgroups. Every archimedean group admits an embedding into the group of real numbers.

## Examples

Archimedean groups include

Non-archimedean groups include