nLab Archimedean ordered local ring

Definition

Since every ordered local ring $R$ has characteristic zero, the positive integers $\mathbb{Z}_+$ are a subset of $R$, with injection $i:\mathbb{Z}_+ \hookrightarrow R$. An Archimedean ordered local ring is an ordered local ring which satisfies the archimedean property: for all elements $a \in R$ and $b \in R$, if $0 \lt a$ and $0 \lt b$, then there exists a positive integer $n \in \mathbb{Z}_+$ such that such that $a \lt i(n) \cdot b$.

Unlike Archimedean ordered fields, which require arithmetic Heyting pretoposes, Archimedean ordered local rings are definable in any arithmetic pretopos.

In analysis

Archimedean ordered local rings are important for modeling notions of infinitesimals. These include the dual numbers, which represent nilsquare infinitesimals and are used to synthetically define differentiable functions in the real numbers, Archimedean ordered Weil rings, which represent nilpotent infintiesimals and are used to synthetically define smooth functions in the real numbers, as well as formal power series on the ground ring, which represent infinitesimals which are not nilpotent and are used to synthetically define analytic functions in the real numbers.

 Kock-Lawvere axiom

An Archimedean ordered local ring $R$ satisfies the Kock-Lawvere axiom if and only if given any Weil $R$-algebra $W$, the canonical function from $W$ to $R^{\mathrm{spec}_R^W}$ the function algebra with domain the formal spectrum of $W$ and codomain $R$ is an $R$-algebra isomorphism.

See also

• Archimedean ordered field
• ordered local ring
• Archimedean ordered Kock field?
• Archimedean ordered Artinian local ring
• Archimedean ordered reduced local ring
• Archimedean ordered local integral domain

 References

• Ieke Moerdijk, Gonzalo E. Reyes: Models for Smooth Infinitesimal Analysis, Springer (1991), doi:10.1007/978-1-4757-4143-8