nLab Archimedean property

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

Let $(A,\lt, +, 0)$ be a strictly ordered cancellative commutative monoid. The positive integers are embedded into the function monoid $A \to A$; there is an injection $inj:\mathbb{N}^+\to (A \to A)$ such that $inj(1) = id_A$ and $inj(s(n)) = inj(n) + id_A$ for all $n:\mathbb{N}^+$.

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

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

Examples

• Archimedean group
• Archimedean ordered field
• Archimedean ordered integral domain
• Archimedean ordered local ring

References

• Wikipedia, Archimedean property