# Archimedean group

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In abstract algebra, a branch of mathematics, an **Archimedean group** is an algebraic structure consisting of a set together with a binary operation and binary relation satisfying certain axioms detailed below. We can also say that an Archimedean group is a linearly ordered group for which the Archimedean property holds. For example, the set **R** of real numbers together with the operation of addition and usual ordering relation (≤) is an Archimedean group. The concept is named after Archimedes.

## Definition

In the subsequent, we use the notation (where is in the set **N** of natural numbers) for the sum of *a* with itself *n* times.

An **Archimedean group** (*G*, +, ≤) is a linearly ordered group subject to the following condition:

for any *a* and *b* in *G* which are greater than *0*, the inequality *na* ≤ *b* holding for every *n* in **N** implies *a* = 0.

## Examples of Archimedean groups

The sets of the integers, the rational numbers, the real numbers, together with the operation of addition and the usual ordering (≤), are Archimedean groups.

## Examples of non-Archimedean groups

An ordered group (*G*, +, ≤) defined as follows is not Archimedean:

*G*=**R**×**R**.- Let
*a*= (*u*,*v*) and*b*= (*x*,*y*) then*a*+*b*= (*u*+*x*,*v*+*y*) *a*≤*b*iff*v*<*y*or (*v*=*y*and*u*≤*x*) (lexicographical order with the least-significant number on the left).

**Proof:** Consider the elements (1, 0) and (0, 1). For all *n* in **N** one evidently has *n* (1, 0) < (0, 1).

For another example, see p-adic number.

## Theorems

For each *a*, *b* in *G* there exist *m*, *n* in **N** such that *ma* ≤ *b* and *a* ≤ *nb*.