Archimedean group
In abstract algebra, a branch of mathematics, an Archimedean group is an algebraic structure. It is a set together with a binary operation and binary relation, with both 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.
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 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). 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 R such that ma ≤ b and a ≤ nb.
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