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.
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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 for any n in N implies a = 0.
The sets of the integers, the rational numbers, the real numbers, together with the operation of addition and the usual ordering (≤), are Archimedean groups.
An ordered group (G, +, ≤) defined as follows is not Archimedean:
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.
For each a, b in G there exist m, n in N such that ma ≤ b and a ≤ nb.