nLab Archimedean property

Contents

 Idea

The Archimedean property states that every positive element in a strictly ordered cancellative commutative monoid is bounded above by a natural number.

So an object satisfying the Archimedean property has no infinite elements. If the strict order is additionally a connected relation and thus a linear order, every infinitesimal element is equal to zero.

Definition

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

Let (A,<,+,0)(A,\lt, +, 0) be a strictly ordered cancellative commutative monoid. The positive integers are embedded into the function monoid 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).

Examples

References

Created on January 9, 2023 at 00:50:55. See the history of this page for a list of all contributions to it.