nLab Archimedean ordered Artinian local ring

Idea

Since every ordered Artinian 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 Artinian local ring is an ordered Artinian 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 $a \lt i(n) \cdot b$.

Archimedean ordered Artinian local rings are important because they are the ordered local rings with nilpotent infinitesimals but no infinite elements or invertible infinitesimals, and thus play an important role in synthetic approaches to analysis and differential geometry.

Archimedean ordered fields are the Archimedean ordered Artinian local rings in which the nilradical is the trivial ideal, or equivalently, in which every non-invertible element is equal to zero.

Properties

Purely real and purely infinitesimal elements

Suppose that $K$ is an Archimedean ordered field and $A$ is an Archimedean ordered Artinian local $K$-algebra. Since $A$ is a local ring, the quotient of $A$ by its ideal of non-invertible elements $D$ is the residue field $K$ itself, and the canonical function used in defining the quotient is the function $\Re:A \to K$ which takes a number $a \in A$ to its purely real component $\Re(a) \in K$. Since $A$ is an ordered $K$-algebra, there is a strictly monotone ring homomorphism $h:K \to A$. An element $a \in A$ is purely real if $h(\Re(a)) = a$, and an element $a \in A$ is purely infinitesimal if it is in the fiber of $\Re$ at $0 \in K$. Zero is the only element in $A$ which is both purely real and purely infinitesimal.

Prelattice structure

Now, suppose that the Archimedean ordered field $K$ has lattice structure $\min:K \times K \to K$ and $\max:K \times K \to K$. Then the Archimedean ordered local $K$-algebra $A$ has a prelattice structure given by functions $(-)\wedge(-):A \times A \to A$ and $(-)\vee(-):A \times A \to A$, defined by

$$a \wedge b \coloneqq h(\min(\Re(a), \Re(b)))$$

and

$$a \vee b \coloneqq h(\max(\Re(a), \Re(b)))$$

Pseudometric and seminorm structure

The distance function on $A$ is given by the function $\rho:A \times A \to K$, defined as

$$\rho(a, b) \coloneqq \max(\Re(a), \Re(b)) - \min(\Re(a), \Re(b))$$

and the absolute value on $A$ is given by the function $\vert-\vert:A \times A \to K$, defined as

$$\vert a \vert \coloneqq \rho(a, 0)$$

The distance function and absolute value are pseudometrics and multiplicative seminorms, because every Archimedean ordered field $K$ embeds into the real numbers $\mathbb{R}$, and since $\min(a, b) \leq \max(a, b)$, the pseudometric and seminorm are always non-negative. This also implies that in every Archimedean ordered field with lattice structure, the pseudometric defined above is a metric.

Smooth and differentiable function structure

Suppose that $K$ is an Archimedean ordered field with lattice structure, and $A$ is an Archimedean ordered Artinian local $K$-algebra. Then continuous functions, differentiable functions, and smooth functions are each definable on $K$ using the algebraic, order, and metric structure on $K$.

The ring homomorphism $h:K \to A$ preserves smooth functions: given a natural number $n \in \mathbb{N}$ and a purely infinitesimal element $\epsilon \in D$ such that $\epsilon^{n + 1} = 0$, then for every smooth function $f \in C^\omega(K)$, there is a function $f_A:A \to A$ such that for all elements $x \in K$, $f_A(h(x)) = h(f(x))$ and

$$f_A(h(x) + \epsilon) = \sum_{i = 0}^{n} \frac{1}{i!} h\left(\frac{d^i f}{d x^i}(x)\right) \epsilon^i$$

If we restrict to Archimedean ordered Artinian local $K$-algebras $A$ where every element of the nilradical $D$ is a nilsquare element, where for all $\epsilon \in D$, $\epsilon^2 = 0$, then the ring homomorphism $h:K \to A$ preserves differentiable functions; for every differentiable function $f:K \to K$ with given derivative $f':K \to K$, there is a function $f_A:A \to A$ such that for all elements $x \in K$ and nilpotent elements $\epsilon \in D$, $f_A(h(x)) = h(f(x))$ and

$$f_A(h(x) + \epsilon) = h(f(x)) + h(f'(x)) \epsilon$$

In particular, every polynomial function $p:K \to K$ lifts to a polynomial function $p_A:A \to A$.

Alternatively, one could use this property to define differentiable and smooth functions in $K$, such as the exponential function, natural logarithm, sine function, and cosine function.

One could also work with partial functions instead. Given a predicate $P$ on the real numbers $\mathbb{R}$, let $I$ denote the set of all elements in $\mathbb{R}$ for which $P$ holds. A partial function $f:\mathbb{R} \to \mathbb{R}$ is equivalently a function $f:I \to \mathbb{R}$ for any such predicate $P$ and set $I$.

A function $f:I \to \mathbb{R}$ is smooth at a subset $S \subseteq I$ with injection $j:S \hookrightarrow \mathbb{R}$ if it has a function $\frac{d^{-} f}{d x^{-}}:\mathbb{N} \times S \to \mathbb{R}$ with $(D^0 j)(a) = a$ for all $a \in S$, such that for all Archimedean ordered Artinian local $\mathbb{R}$-algebras $A$ with ring homomorphism $h_A:\mathbb{R} \to A$, natural numbers $n \in \mathbb{N}$, and purely infinitesimal elements $\epsilon \in D$ such that $\epsilon^{n + 1} = 0$

$$f_A(h_A(j(a)) + \epsilon) = \sum_{i = 0}^{n} \frac{1}{i!} h_A\left(\frac{d^i f}{d x^i}\left(a\right)\right) \epsilon^i$$

A function $f:I \to \mathbb{R}$ is smooth at an element $a \in I$ if it is smooth at the singleton subset $\{a\}$, and a function $f:I \to \mathbb{R}$ is smooth if it is smooth at the improper subset of $I$.

A function $f:I \to \mathbb{R}$ is differentiable at a subset $S \subseteq I$ with injection $j:S \hookrightarrow \mathbb{R}$ if it has a function $\frac{d f}{d x}:S \to \mathbb{R}$ such that for all Archimedean ordered Artinian local $K$-algebras $A$ with ring homomorphism $h:K \to A$ such that for all $\epsilon \in D$, $\epsilon^2 = 0$, for all nilpotent elements $\epsilon \in D$,

$$f_A(h(j(a)) + \epsilon) = h(j(a)) + h\left(\frac{d f}{d x}(a)\right) \epsilon$$

A function $f:I \to \mathbb{R}$ is differentiable at an element $a \in I$ if it is differentiable at the singleton subset $\{a\}$, and a function $f:I \to \mathbb{R}$ is differentiable if it is differentiable at the improper subset of $I$.

Square roots and Euclidean pseudometric structure

Now, assume that $K$ is an Euclidean field as well, in addition to being an Archimedean ordered field. While $K$ has a principal square root function $\sqrt{-}:[0, \infty) \to [0, \infty)$, not every Archimedean ordered local $K$-algebra $A$ has a principal square root function $\sqrt{-}:[0, \infty) \to [0, \infty)$, because purely infinitesimal elements in $A$ are not guaranteed to have square roots. An Archimedean ordered Artinian local $K$-algebra is Euclidean if every nilpotent element has a square root.

However, given an Archimedean ordered Artinian local $K$-algebra $A$, every rank $n$ $A$-module $V$ with basis $v:\mathrm{Fin}(n) \to V$ has a Euclidean pseudometric $\rho_V:V \times V \to K$, given by

$$\rho_V(a, b) \coloneqq \sqrt{\sum_{i \in \mathrm{Fin}(n)} \rho(a_i, b_i)^2}$$

for module elements $a \in V$ and $b \in V$ and scalars $a_i \in A$ and $b_i \in A$ for index $i \in \mathrm{Fin}(n)$, where

$$a = \sum_{i \in \mathrm{Fin}(n)} a_i v_i \quad b = \sum_{i \in \mathrm{Fin}(n)} b_i v_i$$

If $A$ is an ordered field, then the Euclidean pseudometric on $V$ is a metric.

See also

• Archimedean ordered field
• Archimedean ordered local ring
• ordered Artinian local ring