Archimedean group

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 $na$ (where $n$ 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" for any "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".