Archimedean group

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{{ safesubst:#invoke:Unsubst||$N=Unreferenced |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} 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 nab 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)
  • ab iff v < y or (v = y and ux) (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 mab and anb.