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).
Let $(\mathbb{N}^+,1:\mathbb{N}^+,s:\mathbb{N}^+\to \mathbb{N}^+)$ be the set of positive integers.
Let $(A,\lt)$ be a linearly ordered group. The positive integers are embedded into the function group $A \to A$; 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 $a \lt act(n,b)$.
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 $R$-algebra, then it is called an archimedean $R$-algebra, and if it is a field, then it is called an archimedean field.
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.
Every archimedean group is a flat module and a torsion-free group.
Every Dedekind complete archimedean group is isomorphic to the integers, if the group is not dense, or the Dedekind real numbers, if the group is dense.
Archimedean groups include
half integers?
Non-archimedean groups include
Last revised on December 5, 2022 at 02:37:45. See the history of this page for a list of all contributions to it.