nLab Archimedean ordered local ring



Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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 }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)


Formal geometry



Since every ordered local ring RR has characteristic zero, the positive integers +\mathbb{Z}_+ are a subset of RR, with injection i: +Ri:\mathbb{Z}_+ \hookrightarrow R. An Archimedean ordered local ring is an ordered local ring which satisfies the archimedean property: for all elements aRa \in R and bRb \in R, if 0<a0 \lt a and 0<b0 \lt b, then there exists a positive integer n +n \in \mathbb{Z}_+ such that such that a<i(n)ba \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.

See also

Last revised on January 13, 2023 at 08:01:16. See the history of this page for a list of all contributions to it.