transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
symmetric monoidal (∞,1)-category of spectra
Arithmetic for closed intervals of an ordered field.
Let $F$ be an ordered field. The closed intervals of $F$ can be represented by a subset of the Cartesian product set $F \times F$:
The elements $[a, b] \in \mathcal{I}(F)$ of the set are the endpoints of the closed intervals, which we shall call intervals for short.
There is an embedding $i:F \to \mathcal{I}(F)$ defined by
Equality of intervals is defined as
If $R$ has decidable equality, then $\mathcal{I}(R)$ has decidable equality as well.
A positive interval is defined by
A negative interval is defined by
A indefinite interval is defined by
Zero is defined by
Addition is defined by
Negation is defined by
Subtraction is defined by
Intervals do not form an abelian group because $[a, b] - [a, b] = [a - b, b - a]$, and $[a - b, b - a] = 0$ if and only if $a = b$. However, they do form a commutative monoid under zero and addition.
One is defined by
Multiplication is defined by
Intervals form a commutative monoid under one and multiplication. Zero and multiplication also form an absorption monoid. Multiplication does not distribute over addition:
The reciprocal is only defined for intervals that are positive or negative: $0 \lt [a, b]$ or $0 \gt [a, b]$, and is defined by
Division is only defined for intervals in the denominator that are positive or negative: $0 \lt [c, d]$ or $0 \gt [c, d]$, and is defined by
However, division of an interval by itself does not return the constant one: if $0 \lt [a, b]$ or $0 \gt [a, b]$, then
and $[a/b, b/a] = 1$ if and only if $a = b$.
The linear order of the rational intervals is defined as
The opposite linear order of the rational intervals is defined as
The containment relation of the rational intervals is defined as
The opposite containment relation of the rational intervals is defined as
Last revised on May 12, 2022 at 13:48:06. See the history of this page for a list of all contributions to it.