nLab
archimedean group

Contents

Contents

Idea

An archimedean group is a linearly ordered group in which every positive element is bounded above by a natural number.

So an archimedean group has no infinite elements (and thus no non-zero infinitesimal elements).

Definition

Let ( +,1: +,s: + +)(\mathbb{N}^+,1:\mathbb{N}^+,s:\mathbb{N}^+\to \mathbb{N}^+) be the set of positive integers.

Let (A,<)(A,\lt) be a linearly ordered group. The positive integers are embedded into the function group AAA \to A; there is an injection inj: +(AA)inj:\mathbb{N}^+\to (A \to A) such that inj(1)=id Ainj(1) = id_A and inj(s(n))=inj(n)+id Ainj(s(n)) = inj(n) + id_A for all n: +n:\mathbb{N}^+.

The archimedean property states that for every a,b:Aa,b:A such that 0<a0 \lt a and 0<b0 \lt b, then there exist n: +n:\mathbb{N}^+ such that a<inj(n)(b)a \lt inj(n)(b).

By uncurrying injinj one gets an action act:( +×A)Aact: (\mathbb{N}^+\times A) \to A such that act(1,a)=aact(1,a) = a and act(s(n),a)=act(n,a)+aact(s(n),a) = act(n,a) + a for all n: +n:\mathbb{N}^+ and a:Aa:A. The archimedean property then states that for all a,b:Aa,b:A such that 0<a0 \lt a and 0<b0 \lt b, there exist n: +n:\mathbb{N}^+ such that a<act(n,b)a \lt act(n,b).

An archimedean group is a linearly ordered group satisfying the archimedean property.

An archimedean group that is also a linearly ordered ring is called an archimedean ring. If an archimedean ring is a RR-algebra, then it is called an archimedean RR-algebra, and if it is a field, then it is called an archimedean field.

Properties

Every archimedean group is an abelian group and has no bounded cyclic subgroups. Every archimedean group admits an embedding into the group of real numbers.

Examples

Archimedean groups include

Non-archimedean groups include

See also

Last revised on June 15, 2021 at 21:41:06. See the history of this page for a list of all contributions to it.