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 <math>na</math> (where <math>n</math> 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 set of real numbers together with the operation of addition and the usual ordering (≤) is an Archimedean group.
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).
Theorems
For each a, b in G there exist m, n in N such that ma ≤ b and a ≤ nb.
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