# nLab Archimedean ordered local ring

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

#### Algebra

higher algebra

universal algebra

# Contents

## 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 $\mathbb{R}$-algebras 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 $\mathbb{R}$-algebras, which represent nilpotent infintiesimals and are used to synthetically define smooth functions in the real numbers, as well as formal power series on $\mathbb{R}$, which represent infinitesimals which are not nilpotent and are used to synthetically define analytic functions in the real numbers.