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

Examples

Archimedean groups include

Non-archimedean groups include

See also

Last revised on December 5, 2022 at 02:37:45. See the history of this page for a list of all contributions to it.