decimal rational number

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Arithmetic
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A __decimal rational number__ or __decimal fraction__ is a [[rational number]] $r \in \mathbb{Q}$ such that the [[decimal number|decimal]] expansion of $r$ has finitely many digits.

## Definition

### As a subset of the rational numbers

A __decimal rational number__ or __decimal fraction__ is a [[rational number]] $r \in \mathbb{Q}$ such that there exists $n \in \mathbb{N}$ and $a \in \mathbb{Z}$ such that $r = \frac{a}{10^n}$.

### As the localisation of the integers at 10

The [[commutative ring]] of decimal rational numbers $\mathbb{Z}[1/10]$ is the [[localization of a commutative ring|localization]] of the [[integers]] $\mathbb{Z}$ [[localisation of a commutative ring away from an element|away from]] $10$.

### As an initial object of a category

For lack of a better name, let us define a *set with decimal fractions* to be a set $A$ with a function $\iota \in \mathbb{Z} \times \mathbb{N} \to A$, such that

$$\forall a \in \mathbb{Z}. \forall b \in \mathbb{N}. \forall c \in \mathbb{Z}. \forall d \in \mathbb{N}. (a \cdot 10^d = c \cdot 10^b) \implies (\iota(a, b) = \iota(c, d))$$

The integer $a:\mathbb{Z}$ represents the integer if one ignores the decimal separator in the [[decimal number|decimal numeral representation]] of the decimal fraction, and the natural number $b:\mathbb{N}$ represents the number of digits to the left of the final digit where the decimal separator ought to be placed after in the decimal numeral representation of the decimal fraction.  The axiom above is used to state that equivalent decimals are equal: i.e. 1.00 = 1.000 with decimal numeral representations.

A *homomorphism of sets with decimal fractions* between two decimal fraction algebras $A$ and $B$ is a function $f:A \to B$ such that

$$\forall a \in \mathbb{Z}. \forall b \in \mathbb{N}. f(\iota_A(a, b)) = \iota_B(a, b)$$

The *category of sets with decimal fractions* is the category $SwDF$ whose objects $Ob(SwDF)$ are sets with decimal fractions and whose morphisms $Mor(A,B)$ for sets with decimal fractions $A \in Ob(SwDF)$, $B \in Ob(SwDF)$ are homomorphisms of sets with decimal fractions. The set of **decimal fractions**, denoted $\mathbb{Z}[1/10]$, is defined as the initial object in the category of sets with decimal fractions.

### As a higher inductive type ###

In [[homotopy type theory]], the type of **decimal numbers**, denoted $\mathbb{Z}[1/10]$, could be defined as the [[higher inductive type]] generated by:

 * A function $(-)/10^{(-)}:\mathbb{Z} \times \mathbb{N} \rightarrow \mathbb{Z}[1/10]$. The integer $a:\mathbb{Z}$ represents the integer if one ignores the decimal separator in the decimal numeral representation of the decimal number, and the natural number $b:\mathbb{N}$ represents the number of digits to the left of the final digit where the decimal separator ought to be placed after in the decimal numeral representation of the decimal number.
 * A dependent product of functions between identities representing that equivalent decimals are equal:
$$equivdec : \prod_{a:\mathbb{Z}} \prod_{b:\mathbb{N}} \prod_{c:\mathbb{Z}} \prod_{d:\mathbb{N}} (a \cdot 10^d = c \cdot 10^b) \to (a/10^b = c/10^d)$$
(i.e. 1.00 = 1.000 with decimal numeral representations)
 * A set-truncator
$$\tau_0: \prod_{a:\mathbb{Z}[1/10]} \prod_{b:\mathbb{Z}[1/10]} isProp(a=b)$$

## Properties ##

### Decidability ###

+-- {: .num_prop}
###### Proposition
**(Decidability of equality)

For every decimal fraction $p \in \mathbb{Z}[1/10]$ and $q \in \mathbb{Z}[1/10]$, $p = q$ or $\neg(p = q)$.
=--

+-- {: .num_defn}
###### Definition
**(Inequality)

For every decimal fraction $p \in \mathbb{Z}[1/10]$ and $q \in \mathbb{Z}[1/10]$, there is a proposition $p \neq q$ called **inequality** and defined as

$$p \neq q \coloneqq \neg(p = q)$$
=--

+-- {: .num_prop}
###### Proposition
**(Tight apartness relation)

Inequality $\neq$ is a [[tight apartness relation]]: For every decimal fraction $p \in \mathbb{Z}[1/10]$ and $q \in \mathbb{Z}[1/10]$, if $\neg(p \neq q)$, then $p = q$.
=--

### Commutative ring structure ###

+-- {: .num_defn}
###### Definition
**(Addition)

There exists a binary operation $(-)+(-):\mathbb{Z}[1/10] \times \mathbb{Z}[1/10] \to \mathbb{Z}[1/10]$ called **addition** defined as

$$a/10^b + c/10^d \coloneqq (a \cdot 10^d + c \cdot 10^b)/10^{(b + d)}$$

for $a \in \mathbb{Z}$, $b \in \mathbb{N}$, $c \in \mathbb{Z}$, $d \in \mathbb{N}$.
=--

+-- {: .num_prop}
###### Proposition
**(Associativity of addition)

For every decimal fraction $p \in \mathbb{Z}[1/10]$, $q \in \mathbb{Z}[1/10]$, and $r \in \mathbb{Z}[1/10]$, $(p + q) + r = p + (q + r)$.
=--

+-- {: .proof}
###### Proof

By definition of the set of decimal fractions, there exist numbers $a \in \mathbb{Z}$, $b \in \mathbb{N}$, $c \in \mathbb{Z}$, $d \in \mathbb{N}$, $e \in \mathbb{Z}$, $f \in \mathbb{N}$ such that $a/10^b = p$, $c/10^d = q$, and $e/10^f = q$. Thus,

$$p + q = a/10^b + c/10^d = (a \cdot 10^d + c \cdot 10^b)/10^{(b + d)}$$

$$(p + q) + r = (a \cdot 10^d + c \cdot 10^b)/10^{(b + d)} + e/10^f = ((a \cdot 10^d + c \cdot 10^b) \cdot 10^f + e \cdot 10^{(b + d)})/10^{(b + d) + f}$$

$$q + r = c/10^d + e/10^f = (c \cdot 10^f + e \cdot 10^d)/10^{(d + f)}$$

$$p + (q + r) = a/10^b + (c \cdot 10^f + e \cdot 10^d)/10^{(d + f)} = (a \cdot 10^{(d + f)} + (c \cdot 10^f + e \cdot 10^d) \cdot 10^b)/10^{b + (d + f)}$$

by definition of addition. By the distributive property of multiplication of integers

$$((a \cdot 10^d + c \cdot 10^b) \cdot 10^f + e \cdot 10^{(b + d)})/10^{(b + d) + f} = (a \cdot 10^d \cdot 10^f + c \cdot 10^b \cdot 10^f) + e \cdot 10^{(b + d)})/10^{(b + d) + f}$$

$$(a \cdot 10^{(d + f)} + (c \cdot 10^f + e \cdot 10^d) \cdot 10^b)/10^{b + (d + f)} = (a \cdot 10^{(d + f)} + (c \cdot 10^f \cdot 10^b + e \cdot 10^d \cdot 10^b)/10^{b + (d + f)}$$

and by the fact that exponentiation is an $\mathbb{N}$-[[action]] with respect to multiplication and by the commutative property of addition of natural numbers,

$$(a \cdot 10^d \cdot 10^f + c \cdot 10^b \cdot 10^f) + e \cdot 10^{(b + d)})/10^{(b + d) + f} = (a \cdot 10^{d + f} + c \cdot 10^{b + f}) + e \cdot 10^{(b + d)})/10^{(b + d) + f}$$

$$(a \cdot 10^{(d + f)} + (c \cdot 10^f \cdot 10^b + e \cdot 10^d \cdot 10^b)/10^{b + (d + f)} = (a \cdot 10^{d + f} + (c \cdot 10^{b + f} + e \cdot 10^{b + d})/10^{b + (d + f)}$$

Finally, by the associative property of addition of integers and natural numbers,

$$(a \cdot 10^{d + f} + c \cdot 10^{b + f}) + e \cdot 10^{(b + d)})/10^{(b + d) + f} = (a \cdot 10^{d + f} + (c \cdot 10^{b + f} + e \cdot 10^{b + d})/10^{b + (d + f)}$$

Thus, $(p + q) + r = p + (q + r)$.

=--

+-- {: .num_prop}
###### Proposition
**(Commutativity of addition)

For every decimal fraction $p \in \mathbb{Z}[1/10]$ and $q \in \mathbb{Z}[1/10]$, $p + q = q + p$.
=--

+-- {: .proof}
###### Proof

By definition of the set of decimal fractions, there exist numbers $a \in \mathbb{Z}$, $b \in \mathbb{N}$, $c \in \mathbb{Z}$, $d \in \mathbb{N}$ such that $a/10^b = p$ and $c/10^d = q$. Thus,

$$p + q = a/10^b + c/10^d = (a \cdot 10^d + c \cdot 10^b)/10^{(b + d)}$$

and

$$q + p = c/10^d + a/10^b = (c \cdot 10^b + a \cdot 10^d)/10^{(d + b)}$$

by definition of addition. By the commutativity of addition in the integers and the natural numbers,

$$(a \cdot 10^d + c \cdot 10^b)/10^{(b + d)} = (c \cdot 10^b + a \cdot 10^d)/10^{(d + b)}$$

Thus, $p + q = q + p$.

=--

+-- {: .num_defn}
###### Definition
**(Zero)

There exists an element $0 \in \mathbb{Z}[1/10]$ called **zero** and defined as $0 \coloneqq 0/10^0$.
=--

+-- {: .num_prop}
###### Proposition
**(Zero is a left identity of addition)

For every decimal fraction $p \in \mathbb{Z}[1/10]$, $0 + p = p$.
=--

+-- {: .proof}
###### Proof

By definition of the set of decimal fractions, there exist numbers $a \in \mathbb{Z}$, $b \in \mathbb{N}$ such that $a/10^b = p$. Thus,

$$0 + p = 0/10^0 + a/10^b = (0 \cdot 10^b + a \cdot 10^0)/10^{(0 + b)}$$

by definition of addition. By the absorption property of $0$ with respect to multiplication in the integers,

$$(0 \cdot 10^b + a \cdot 10^0)/10^{(0 + b)} = (a \cdot 10^0)/10^{(0 + b)}$$

by the fact that exponentiation is an $\mathbb{N}$-[[action]] with respect to multiplication in the integers,

$$(a \cdot 10^0)/10^{(0 + b)} = (a \cdot 1)/10^{(0 + b)}$$

by the fact that 1 is a right identity element with respect to multiplication in the integers,

$$(a \cdot 1)/10^{(0 + b)} = a/10^{(0 + b)}$$

and by the fact that 0 is a left identity element with respect to addition in the natural numbers,

$$a/10^{(0 + b)} = a/10^{b}$$

Thus, $0 + p = p$.

=--

+-- {: .num_prop}
###### Proposition
**(Zero is a right identity of addition)

For every decimal fraction $p \in \mathbb{Z}[1/10]$, $p + 0 = p$.
=--

+-- {: .proof}
###### Proof

By definition of the set of decimal fractions, there exist numbers $a \in \mathbb{Z}$, $b \in \mathbb{N}$ such that $a/10^b = p$. Thus,

$$p + 0 = a/10^b + 0/10^0 = (a \cdot 10^0 + 0 \cdot 10^b)/10^{(b + 0)}$$

by definition of addition. By the absorption property of $0$ with respect to multiplication in the integers,

$$(a \cdot 10^0 + 0 \cdot 10^b)/10^{(b + 0)} = (a \cdot 10^0)/10^{(b + 0)}$$

by the fact that exponentiation is an $\mathbb{N}$-[[action]] with respect to multiplication in the integers,

$$(a \cdot 10^0)/10^{(b + 0)} = (a \cdot 1)/10^{(b + 0)}$$

by the fact that 1 is a right identity element with respect to multiplication in the integers,

$$(a \cdot 1)/10^{(b + 0)} = a/10^{(b + 0)}$$

and by that 0 is a left identity element with respect to addition in the natural numbers,

$$a/10^{(b + 0)} = a/10^{b}$$

Thus, $p + 0 = p$.

=--

+-- {: .num_defn}
###### Definition
**(Negation)

There exists a function $-(-):\mathbb{Z}[1/10] \to \mathbb{Z}[1/10]$ called **negation** and defined as

$$-(a/10^b) \coloneqq (-a)/10^b$$

for $a \in \mathbb{Z}$, $b \in \mathbb{N}$.
=--

+-- {: .num_prop}
###### Proposition
**(Negation is a left inverse of addition)

For every decimal fraction $p \in \mathbb{Z}[1/10]$, $-p + p = 0$.
=--

+-- {: .proof}
###### Proof

By definition of the set of decimal fractions, there exist numbers $a \in \mathbb{Z}$, $b \in \mathbb{N}$ such that $a/10^b = p$. Thus,

$$-p + p = (-a)/10^b + a/10^0 = ((-a) \cdot 10^b + a \cdot 10^b)/10^{(b + b)}$$

by definition of addition. By the fact that multiplication is a [[bilinear function]] in the integers,

$$((-a) \cdot 10^b + a \cdot 10^b)/10^{(b + b)} = (-(a \cdot 10^b) + a \cdot 10^b)/10^{(b + b)}$$

and by the fact that negation is a left inverse of addition in the integers,

$$(-(a \cdot 10^b) + a \cdot 10^b)/10^{(b + b)}/10^{(b + b)} = 0/10^{(b + b)}$$

and by definition of equality of decimal fractions, since

$$0 \cdot 10^{(b + b)} = 0 \cdot 10^0$$

this means that

$$0/10^{(b + b)} = 0/10^0$$

Thus, $-p + p = 0$.

=--

+-- {: .num_prop}
###### Proposition
**(Negation is a right inverse of addition)

For every decimal fraction $p \in \mathbb{Z}[1/10]$, $p + (-p) = 0$.
=--

+-- {: .proof}
###### Proof

By definition of the set of decimal fractions, there exist numbers $a \in \mathbb{Z}$, $b \in \mathbb{N}$ such that $a/10^b = p$. Thus,

$$p + -p = a/10^b + (-a)/10^0 = (a \cdot 10^b + (-a) \cdot 10^b)/10^{(b + b)}$$

by definition of addition. By the fact that multiplication is a [[bilinear function]] in the integers,

$$(a \cdot 10^b + (-a) \cdot 10^b)/10^{(b + b)} = (a \cdot 10^b + -(a \cdot 10^b))/10^{(b + b)}$$

and by the fact that negation is a right inverse of addition in the integers,

$$(a \cdot 10^b + -(a \cdot 10^b))/10^{(b + b)} = 0/10^{(b + b)}$$

and by definition of equality of decimal fractions, since

$$0 \cdot 10^{(b + b)} = 0 \cdot 10^0$$

this means that

$$0/10^{(b + b)} = 0/10^0$$

Thus, $p + -p = 0$.

=--

+-- {: .num_defn}
###### Definition
**(Subtraction)

There exists a binary operation $(-)-(-):\mathbb{Z}[1/10] \times \mathbb{Z}[1/10] \to \mathbb{Z}[1/10]$ called **subtraction** defined as

$$a/10^b - c/10^d \coloneqq (a \cdot 10^d - c \cdot 10^b)/10^{(b + d)}$$

for $a \in \mathbb{Z}$, $b \in \mathbb{N}$, $c \in \mathbb{Z}$, $d \in \mathbb{N}$.
=--

+-- {: .num_prop}
###### Proposition
**(Abelian group)

The decimal fractions $(\mathbb{Z}[1/10], 0, +, -)$ form an [[abelian group]].
=--

+-- {: .num_defn}
###### Definition
**(Multiplication)

There exists a binary operation $(-)\cdot (-):\mathbb{Z}[1/10] \times \mathbb{Z}[1/10] \to \mathbb{Z}[1/10]$ called **multiplication** defined as

$$a/10^b \cdot c/10^d \coloneqq (a \cdot c)/10^{(b + d)}$$

for $a \in \mathbb{Z}$, $b \in \mathbb{N}$, $c \in \mathbb{Z}$, $d \in \mathbb{N}$.
=--

+-- {: .num_prop}
###### Proposition
**(Left distributivity of multiplication over addition)

For every decimal fraction $p \in \mathbb{Z}[1/10]$, $q \in \mathbb{Z}[1/10]$, and $r \in \mathbb{Z}[1/10]$, $p \cdot (q + r) = p \cdot q + p \cdot r$.
=--

+-- {: .proof}
###### Proof

By definition of the set of decimal fractions, there exist numbers $a \in \mathbb{Z}$, $b \in \mathbb{N}$, $c \in \mathbb{Z}$, $d \in \mathbb{N}$, $e \in \mathbb{Z}$, $f \in \mathbb{N}$ such that $a/10^b = p$, $c/10^d = q$, and $e/10^f = q$. Thus,

$$q + r = c/10^d + e/10^f = (c \cdot 10^f + e \cdot 10^d)/10^{(d + f)}$$

$$p \cdot (q + r) = a/10^b \cdot (c \cdot 10^f + e \cdot 10^d)/10^{(d + f)} = (a \cdot (c \cdot 10^f + e \cdot 10^d))/10^{b + (d + f)}$$

$$p \cdot q = a/10^b \cdot c/10^d = (a \cdot c)/10^{(b + d)}$$

$$p \cdot r = a/10^b \cdot e/10^f = (a \cdot e)/10^{(b + f)}$$

$$p \cdot q + p \cdot r = (a \cdot c)/10^{(b + d)} + (a \cdot e)/10^{(b + f)} = ((a \cdot c) \cdot 10^{(b + f)} + (a \cdot e) \cdot 10^{(b + d)})/10^{(b + d) + (b + f)}$$

by definition of addition and multiplication. By left distributivity of multiplication over addition of integers,

$$(a \cdot (c \cdot 10^f + e \cdot 10^d))/10^{b + (d + f)} = (a \cdot (c \cdot 10^f) + a \cdot (e \cdot 10^d))/10^{b + (d + f)}$$

and by associativity of multiplication of integers,

$$(a \cdot (c \cdot 10^f) + a \cdot (e \cdot 10^d))/10^{b + (d + f)} = ((a \cdot c) \cdot 10^f + (a \cdot e) \cdot 10^d)/10^{b + (d + f)}$$

Now we consider the following integer expressions: by the right distributive property of multiplication over addition in integers,

$$((a \cdot c) \cdot 10^f + (a \cdot e) \cdot 10^d) \cdot 10^{(b + d) + (b + f)} = ((a \cdot c) \cdot 10^f) \cdot 10^{(b + d) + (b + f)} + ((a \cdot e) \cdot 10^d) \cdot 10^{(b + d) + (b + f)}$$

$$((a \cdot c) \cdot 10^{(b + f)} + (a \cdot e) \cdot 10^{(b + d)}) \cdot 10^{b + (d + f)} = ((a \cdot c) \cdot 10^{(b + f)}) \cdot 10^{b + (d + f)} + ((a \cdot e) \cdot 10^{(b + d)}) \cdot 10^{b + (d + f)}$$

and by the associative property of multiplication in integers,

$$((a \cdot c) \cdot 10^f) \cdot 10^{(b + d) + (b + f)} + ((a \cdot e) \cdot 10^d) \cdot 10^{(b + d) + (b + f)} = (a \cdot c) \cdot (10^f \cdot 10^{(b + d) + (b + f)}) + (a \cdot e) \cdot (10^d \cdot 10^{(b + d) + (b + f)})$$

$$((a \cdot c) \cdot 10^{(b + f)}) \cdot 10^{b + (d + f)} + ((a \cdot e) \cdot 10^{(b + d)}) \cdot 10^{b + (d + f)} = (a \cdot c) \cdot (10^{(b + f)} \cdot 10^{b + (d + f)}) + (a \cdot e) \cdot (10^{(b + d)} \cdot 10^{b + (d + f)})$$

By the fact that exponentiation is an $\mathbb{N}$-action over multiplication on the integers,

$$(a \cdot c) \cdot (10^f \cdot 10^{(b + d) + (b + f)}) + (a \cdot e) \cdot (10^d \cdot 10^{(b + d) + (b + f)}) = (a \cdot c) \cdot 10^{f + ((b + d) + (b + f))} + (a \cdot e) \cdot (10^{d + ((b + d) + (b + f))}$$

$$(a \cdot c) \cdot (10^{(b + f)} \cdot 10^{b + (d + f)}) + (a \cdot e) \cdot (10^{(b + d)} \cdot 10^{b + (d + f)}) = (a \cdot c) \cdot 10^{(b + f) + (b + (d + f))} + (a \cdot e) \cdot 10^{(b + d) + (b + (d + f))}$$

and by the associativity and commutativity of addition on the natural numbers,

$$(a \cdot c) \cdot 10^{f + ((b + d) + (b + f))} + (a \cdot e) \cdot (10^{d + ((b + d) + (b + f))} = (a \cdot c) \cdot 10^{(f + (b + d)) + (b + f)} + (a \cdot e) \cdot (10^{d + ((b + f) + (b + d))}$$

$$(a \cdot c) \cdot 10^{(f + (b + d)) + (b + f)} + (a \cdot e) \cdot (10^{d + ((b + f) + (b + d))} = (a \cdot c) \cdot 10^{(b + f) + ((b + d) + f)} + (a \cdot e) \cdot (10^{(d + (b + f)) + (b + d)}$$

$$(a \cdot c) \cdot 10^{(b + f) + ((b + d) + f)} + (a \cdot e) \cdot (10^{(d + (b + f)) + (b + d)} = (a \cdot c) \cdot 10^{(b + f) + (b + (d + f))} + (a \cdot e) \cdot (10^{(b + d) + ((b + f) + d)}$$

$$(a \cdot c) \cdot 10^{(b + f) + (b + (d + f))} + (a \cdot e) \cdot (10^{(b + d) + ((b + f) + d)} = (a \cdot c) \cdot 10^{(b + f) + (b + (d + f))} + (a \cdot e) \cdot (10^{(b + d) + (b + (f + d))}$$

$$(a \cdot c) \cdot 10^{(b + f) + (b + (d + f))} + (a \cdot e) \cdot (10^{(b + d) + (b + (f + d))} = (a \cdot c) \cdot 10^{(b + f) + (b + (d + f))} + (a \cdot e) \cdot (10^{(b + d) + (b + (d + f))}$$

Thus,

$$((a \cdot c) \cdot 10^f + (a \cdot e) \cdot 10^d) \cdot 10^{(b + d) + (b + f)} = ((a \cdot c) \cdot 10^{(b + f)} + (a \cdot e) \cdot 10^{(b + d)}) \cdot 10^{b + (d + f)}$$

By definition of equality of decimal fractions, since

$$((a \cdot c) \cdot 10^f + (a \cdot e) \cdot 10^d) \cdot 10^{(b + d) + (b + f)} = ((a \cdot c) \cdot 10^{(b + f)} + (a \cdot e) \cdot 10^{(b + d)}) \cdot 10^{b + (d + f)}$$

then

$$((a \cdot c) \cdot 10^f) + (a \cdot e) \cdot 10^d))/10^{b + (d + f)} = ((a \cdot c) \cdot 10^{(b + f)} + (a \cdot e) \cdot 10^{(b + d)})/10^{(b + d) + (b + f)}$$

Thus, $p \cdot (q + r) = p \cdot q + p \cdot r$.

=--

+-- {: .num_prop}
###### Proposition
**(Right distributivity of multiplication over addition)

For every decimal fraction $p \in \mathbb{Z}[1/10]$, $q \in \mathbb{Z}[1/10]$, and $r \in \mathbb{Z}[1/10]$, $(p + q) \cdot r = p \cdot r + q \cdot r$.
=--

+-- {: .proof}
###### Proof

By definition of the set of decimal fractions, there exist numbers $a \in \mathbb{Z}$, $b \in \mathbb{N}$, $c \in \mathbb{Z}$, $d \in \mathbb{N}$, $e \in \mathbb{Z}$, $f \in \mathbb{N}$ such that $a/10^b = p$, $c/10^d = q$, and $e/10^f = q$. Thus,

$$p + q = a/10^b + c/10^d = (a \cdot 10^d + c \cdot 10^b)/10^{(b + d)}$$

$$(p + q) \cdot r = (a \cdot 10^d + c \cdot 10^b)/10^{(b + d)} \cdot e/10^f = ((a \cdot 10^d + c \cdot 10^b) \cdot e)/10^{(b + d) + f)}$$

$$p \cdot r = a/10^b \cdot e/10^f = (a \cdot e)/10^{(b + f)}$$

$$q \cdot r = c/10^d \cdot c/10^d = (c \cdot e)/10^{(d + f)}$$

$$p \cdot r + q \cdot r = (a \cdot e)/10^{(b + f)} + (c \cdot e)/10^{(d + f)} = ((a \cdot e) \cdot 10^{(d + f)} + (c \cdot e) \cdot 10^{(b + f)})/10^{(b + f) + (d + f)}$$

by definition of addition and multiplication. By right distributivity of multiplication over addition of integers,

$$((a \cdot 10^d + c \cdot 10^b) \cdot e)/10^{(b + d) + f)} = ((a \cdot 10^d) \cdot e + (c \cdot 10^b) \cdot e)/10^{(b + d) + f)}$$

and by associativity of multiplication of integers,

$$((a \cdot 10^d) \cdot e + (c \cdot 10^b) \cdot e)/10^{(b + d) + f)} = (a \cdot (10^d \cdot e) + c \cdot (10^b \cdot e))/10^{(b + d) + f)}$$

and by commutativity of multiplication of integers,

$$(a \cdot (10^d \cdot e) + c \cdot (10^b \cdot e))/10^{(b + d) + f)} = (a \cdot (e \cdot 10^d) + c \cdot (e \cdot 10^b))/10^{(b + d) + f)}$$

and by associativity of multiplication of integers,

$$(a \cdot (e \cdot 10^d) + c \cdot (e \cdot 10^b))/10^{(b + d) + f)} = ((a \cdot e) \cdot 10^d + (c \cdot e) \cdot 10^b)/10^{(b + d) + f)}$$

Now we consider the following integer expressions: by the right distributive property of multiplication over addition in integers,

$$((a \cdot e) \cdot 10^d + (c \cdot e) \cdot 10^b) \cdot 10^{(b + f) + (d + f)} = ((a \cdot e) \cdot 10^d) \cdot 10^{(b + f) + (d + f)} + ((c \cdot e) \cdot 10^b) \cdot 10^{(b + f) + (d + f)}$$

$$((a \cdot e) \cdot 10^{(d + f)} + (c \cdot e) \cdot 10^{(b + f)}) \cdot 10^{(b + d) + f)} = ((a \cdot e) \cdot 10^{(d + f)}) \cdot 10^{(b + d) + f)} + ((c \cdot e) \cdot 10^{(b + f)}) \cdot 10^{(b + d) + f)}$$

and by the associative property of multiplication in integers,

$$((a \cdot e) \cdot 10^d) \cdot 10^{(b + f) + (d + f)} + ((c \cdot e) \cdot 10^b) \cdot 10^{(b + f) + (d + f)} = (a \cdot e) \cdot (10^d \cdot 10^{(b + f) + (d + f)}) + (c \cdot e) \cdot (10^b \cdot 10^{(b + f) + (d + f)})$$

$$((a \cdot e) \cdot 10^{(d + f)}) \cdot 10^{(b + d) + f)} + (c \cdot e) \cdot 10^{(b + f)}) \cdot 10^{(b + d) + f)} = (a \cdot e) \cdot (10^{(d + f)} \cdot 10^{(b + d) + f)}) + (c \cdot e) \cdot (10^{(b + f)} \cdot 10^{(b + d) + f)})$$

By the fact that exponentiation is an $\mathbb{N}$-action over multiplication on the integers,

$$(a \cdot e) \cdot (10^d \cdot 10^{(b + f) + (d + f)}) + (c \cdot e) \cdot (10^b \cdot 10^{(b + f) + (d + f)}) = (a \cdot e) \cdot 10^{d + ((b + f) + (d + f))} + (c \cdot e) \cdot (10^{b + ((b + f) + (d + f))}$$

$$(a \cdot e) \cdot (10^{(d + f)} \cdot 10^{(b + d) + f)}) + (c \cdot e) \cdot (10^{(b + f)} \cdot 10^{(b + d) + f)}) = (a \cdot e) \cdot 10^{(d + f) + ((b + d) + f)} + (c \cdot e) \cdot 10^{(b + f) + ((b + d) + f)}$$

and by the associativity and commutativity of addition on the natural numbers,

$$(a \cdot e) \cdot 10^{d + ((b + f) + (d + f))} + (c \cdot e) \cdot (10^{b + ((b + f) + (d + f))} = (a \cdot e) \cdot 10^{(d + (b + f)) + (d + f)} + (c \cdot e) \cdot (10^{b + ((d + f) + (b + f))}$$

$$(a \cdot e) \cdot 10^{(d + (b + f)) + (d + f)} + (c \cdot e) \cdot (10^{b + ((d + f) + (b + f))} = (a \cdot e) \cdot 10^{(d + f) + (d + (b + f))} + (c \cdot e) \cdot (10^{(b + (d + f)) + (b + f)}$$

$$(a \cdot e) \cdot 10^{(d + (b + f)) + (d + f)} + (c \cdot e) \cdot (10^{b + ((d + f) + (b + f))} = (a \cdot e) \cdot 10^{(d + f) + ((d + b) + f)} + (c \cdot e) \cdot (10^{(b + f) + (b + (d + f))}$$

$$(a \cdot e) \cdot 10^{(d + f) + ((d + b) + f)} + (c \cdot e) \cdot (10^{(b + f) + (b + (d + f))} = (a \cdot e) \cdot 10^{(d + f) + ((b +d) + f)} + (c \cdot e) \cdot (10^{(b + f) + ((b + d) + f)}$$

Thus,

$$((a \cdot e) \cdot 10^d + (c \cdot e) \cdot 10^b) \cdot 10^{(b + f) + (d + f)} = ((a \cdot e) \cdot 10^{(d + f)} + (c \cdot e) \cdot 10^{(b + f)}) \cdot 10^{(b + d) + f)}$$

By definition of equality of decimal fractions, since

$$((a \cdot e) \cdot 10^d + (c \cdot e) \cdot 10^b) \cdot 10^{(b + f) + (d + f)} = ((a \cdot e) \cdot 10^{(d + f)} + (c \cdot e) \cdot 10^{(b + f)}) \cdot 10^{(b + d) + f)}$$

then

$$((a \cdot e) \cdot 10^d + (c \cdot e) \cdot 10^b)/10^{(b + d) + f)} = ((a \cdot e) \cdot 10^{(d + f)} + (c \cdot e) \cdot 10^{(b + f)})/10^{(b + f) + (d + f)}$$

Thus, $(p + q) \cdot r = p \cdot r + q \cdot r$.

=--

+-- {: .num_prop}
###### Proposition
**(Associativity of multiplication)

For every decimal fraction $p \in \mathbb{Z}[1/10]$, $q \in \mathbb{Z}[1/10]$, and $r \in \mathbb{Z}[1/10]$, $(p \cdot q) \cdot r = p \cdot (q \cdot r)$.
=--

+-- {: .proof}
###### Proof

By definition of the set of decimal fractions, there exist numbers $a \in \mathbb{Z}$, $b \in \mathbb{N}$, $c \in \mathbb{Z}$, $d \in \mathbb{N}$, $e \in \mathbb{Z}$, $f \in \mathbb{N}$ such that $a/10^b = p$, $c/10^d = q$, and $e/10^f = q$. Thus,

$$p \cdot q = a/10^b \cdot c/10^d = (a \cdot c)/10^{(b + d)}$$

$$(p \cdot q) \cdot r = (a \cdot c)/10^{(b + d)} + e/10^f = ((a \cdot c) \cdot e)/10^{(b + d) + f}$$

$$q \cdot r = c/10^d \cdot e/10^f = (c \cdot e)/10^{(d + f)}$$

$$p \cdot (q \cdot r) = a/10^b + (c \cdot e)/10^{(d + f)} = (a \cdot (c \cdot e))/10^{b + (d + f)}$$

by definition of addition. By the associative property of multiplication of integers and addition of natural numbers,

$$((a \cdot c) \cdot e)/10^{(b + d) + f} = (a \cdot (c \cdot e))/10^{b + (d + f)}$$

Thus, $(p \cdot q) \cdot r = p \cdot (q \cdot r)$.

=--

+-- {: .num_prop}
###### Proposition
**(Commutativity of multiplication)

For every decimal fraction $p \in \mathbb{Z}[1/10]$ and $q \in \mathbb{Z}[1/10]$, $p \cdot q = q \cdot p$.
=--

+-- {: .proof}
###### Proof

By definition of the set of decimal fractions, there exist numbers $a \in \mathbb{Z}$, $b \in \mathbb{N}$, $c \in \mathbb{Z}$, $d \in \mathbb{N}$ such that $a/10^b = p$ and $c/10^d = q$. Thus,

$$p \cdot q = a/10^b \cdot c/10^d = (a \cdot c)/10^{(b + d)}$$

and

$$q \cdot p = c/10^d \cdot a/10^b = (c \cdot a)/10^{(d + b)}$$

by definition of addition. By the commutativity of multiplication in the integers and the commutativity of addition in the natural numbers,

$$(a \cdot c)/10^{(b + d)} = (c \cdot a)/10^{(d + b)}$$

Thus, $p \cdot q = q \cdot p$.

=--

+-- {: .num_defn}
###### Definition
**(One)

There exists an element $1 \in \mathbb{Z}[1/10]$ called **one** and defined as $1 \coloneqq 1/10^0$.
=--

+-- {: .num_prop}
###### Proposition
**(One is a left identity of multiplication)

For every decimal fraction $p \in \mathbb{Z}[1/10]$, $1 \cdot p = p$.
=--

+-- {: .proof}
###### Proof

By definition of the set of decimal fractions, there exist numbers $a \in \mathbb{Z}$, $b \in \mathbb{N}$ such that $a/10^b = p$. Thus,

$$1 \cdot p = 1/10^0 \cdot a/10^b = (1 \cdot a)/10^{(0 + b)}$$

by the fact that 1 is a left identity element with respect to multiplication in the integers,

$$(1 \cdot a)/10^{(0 + b)} = a/10^{(0 + b)}$$

and by the fact that 0 is a left identity element with respect to addition in the natural numbers,

$$a/10^{(0 + b)} = a/10^{b}$$

Thus, $1 \cdot p = p$.

=--

+-- {: .num_prop}
###### Proposition
**(One is a right identity of multiplication)

For every decimal fraction $p \in \mathbb{Z}[1/10]$, $p \cdot 1 = p$.
=--

+-- {: .proof}
###### Proof

By definition of the set of decimal fractions, there exist numbers $a \in \mathbb{Z}$, $b \in \mathbb{N}$ such that $a/10^b = p$. Thus,

$$p \cdot 1 = a/10^b \cdot 1/10^0 = (a \cdot 1)/10^{(b + 0)}$$

by the fact that 1 is a right identity element with respect to multiplication in the integers,

$$(a \cdot 1)/10^{(b + 0)} = a/10^{(b + 0)}$$

and by the fact that 0 is a right identity element with respect to addition in the natural numbers,

$$a/10^{(b + 0)} = a/10^{b}$$

Thus, $p \cdot 1 = p$.

=--

+-- {: .num_prop}
###### Proposition
**(Commutative ring)

The decimal fractions $(\mathbb{Z}[1/10], 0, +, -, 1, \cdot)$ form a [[commutative ring]].
=--

+-- {: .num_defn}
###### Definition
**(Exponentiation)

There exists a binary operation $(-)^{(-)}:\mathbb{Z}[1/10] \times \mathbb{N} \to \mathbb{Z}[1/10]$ called **exponentiation** defined as

$$(a/10^b)^n \coloneqq (a^n)/10^{(b \cdot n)}$$

for $a \in \mathbb{Z}$, $b \in \mathbb{N}$, $n \in \mathbb{N}$.
=--

+-- {: .num_prop}
###### Proposition
**(10 is a unit)

There is an element $1/10 \in \mathbb{Z}[1/10]$ such that $1/10 \cdot 10 = 1$ and $10 \cdot 1/10 = 1$
=--

+-- {: .num_prop}
###### Proposition
**(Powers of 10 are isomorphic to the integers)

There is an injection $10^{(-)}: \mathbb{Z} \hookrightarrows \mathbb{Z}[1/10]$.
=--

### Order

+-- {: .num_prop}
###### Definition
**(Positive numbers)

Given $a \in \mathbb{Z}$ and $b \in \mathbb{N}$, a decimal fraction $a/10^b \in \mathbb{Z}[1/10]$ **is positive** if $a \gt 0$: i.e. the predicate $\mathrm{isPositive}(a/10^b)$ is defined as

$$\mathrm{isPositive}(a/10^b) \coloneqq a \gt 0$$
=--

+-- {: .num_prop}
###### Proposition
**(Decidability of positive numbers)

Given $a \in \mathbb{Z}$, $b \in \mathbb{N}$, the predicate $\mathrm{isPositive}(a/10^b)$ is a [[decidable proposition]].
=--

+-- {: .proof}
###### Proof
By definition, $\mathrm{isPositive}(a/10^b)$ is the same as $a \gt 0$. Because the relation $a \gt 0$ is decidable for every integer $a \in \mathbb{Z}$, $\mathrm{isPositive}(a/10^b)$ is decidable for every integer $a \in \mathbb{Z}$ and natural number $b \in \mathbb{N}$.

=--

+-- {: .num_defn}
###### Definition
**(Less than)

For every decimal fraction $p \in \mathbb{Z}[1/10]$ and $q \in \mathbb{Z}[1/10]$, there is a proposition $p \lt q$ called **less than** and defined as

$$p \lt q \coloneqq \mathrm{isPositive}(q - p)$$
=--

+-- {: .num_prop}
###### Proposition
**(Decidability of less than)

Given $p \in \mathbb{Z}[1/10]$ and $q \in \mathbb{Z}[1/10]$, the predicate $p \lt q$ is a [[decidable proposition]].
=--

+-- {: .proof}
###### Proof
By definition, $p \lt q$ is the same as $\mathrm{isPositive}(q - p)$. Because the relation $\mathrm{isPositive}(p)$ is decidable for every decimal fraction $p \in \mathbb{Z}[1/10]$, $\mathrm{isPositive}(q - p)$ and $p \lt q$ is decidable for every decimal fraction $p \in \mathbb{Z}[1/10]$ and $q \in \mathbb{Z}[1/10]$.

=--

+-- {: .num_prop}
###### Definition
**(Negative numbers)

Given $a \in \mathbb{Z}$ and $b \in \mathbb{N}$, a decimal fraction $a/10^b \in \mathbb{Z}[1/10]$ **is negative** if $a \lt 0$; i.e. the predicate $\mathrm{isNegative}(a/10^b)$ is defined as

$$\mathrm{isNegative}(a/10^b) \coloneqq a \lt 0$$
=--

+-- {: .num_prop}
###### Proposition
**(Decidability of negative numbers)

Given $a \in \mathbb{Z}$, $b \in \mathbb{N}$, the predicate $\mathrm{isNegative}(a/10^b)$ is a [[decidable proposition]].
=--

+-- {: .proof}
###### Proof
By definition, $\mathrm{isNegative}(a/10^b)$ is the same as $a \lt 0$. Because the relation $a \lt 0$ is decidable for every integer $a \in \mathbb{Z}$, $\mathrm{isNegative}(a/10^b)$ is decidable for every integer $a \in \mathbb{Z}$ and natural number $b \in \mathbb{N}$

=--

+-- {: .num_defn}
###### Definition
**(Greater than)

For every decimal fraction $p \in \mathbb{Z}[1/10]$ and $q \in \mathbb{Z}[1/10]$, there is a proposition $p \gt q$ called **greater than** and defined as

$$p \lt q \coloneqq \mathrm{isNegative}(q - p)$$
=--

+-- {: .num_prop}
###### Proposition
**(Decidability of greater than)

Given $p \in \mathbb{Z}[1/10]$ and $q \in \mathbb{Z}[1/10]$, the predicate $p \gt q$ is a [[decidable proposition]].
=--

+-- {: .proof}
###### Proof
By definition, $p \gt q$ is the same as $\mathrm{isNegative}(q - p)$. Because the relation $\mathrm{isNegative}(p)$ is decidable for every decimal fraction $p \in \mathbb{Z}[1/10]$, $\mathrm{isNegative}(q - p)$ and $p \gt q$ is decidable for every decimal fraction $p \in \mathbb{Z}[1/10]$ and $q \in \mathbb{Z}[1/10]$.
=--

+-- {: .num_prop}
###### Definition
**(Non-negative numbers)

Given $a \in \mathbb{Z}$ and $b \in \mathbb{N}$, a decimal fraction $a/10^b \in \mathbb{Z}[1/10]$ **is non-negative** if $a \geq 0$; i.e. the predicate $\mathrm{isNonNegative}(a/10^b)$ is defined as

$$\mathrm{isNonNegative}(a/10^b) \coloneqq a \geq 0$$
=--

+-- {: .num_prop}
###### Proposition
**(Decidability of non-negative numbers)

Given $a \in \mathbb{Z}$, $b \in \mathbb{N}$, the predicate $\mathrm{isNonNegative}(a/10^b)$ is a [[decidable proposition]].
=--

+-- {: .proof}
###### Proof

By definition, $\mathrm{isNonNegative}(a/10^b)$ is the same as $a \geq 0$. Because the relation $a \geq 0$ is decidable for every integer $a \in \mathbb{Z}$, $\mathrm{isNonNegative}(a/10^b)$ is decidable for every integer $a \in \mathbb{Z}$ and natural number $b \in \mathbb{N}$
=--

+-- {: .num_defn}
###### Definition
**(Less than or equal to)

For every decimal fraction $p \in \mathbb{Z}[1/10]$ and $q \in \mathbb{Z}[1/10]$, there is a proposition $p \leq q$ called **less than or equal to** and defined as

$$p \leq q \coloneqq \mathrm{isNonNegative}(q - p)$$
=--

+-- {: .num_prop}
###### Proposition
**(Decidability of less than or equal to)

Given $p \in \mathbb{Z}[1/10]$ and $q \in \mathbb{Z}[1/10]$, the predicate $p \leq q$ is a [[decidable proposition]].
=--

+-- {: .proof}
###### Proof
By definition, $p \leq q$ is the same as $\mathrm{isNonNegative}(q - p)$. Because the relation $\mathrm{isNonNegative}(p)$ is decidable for every decimal fraction $p \in \mathbb{Z}[1/10]$, $\mathrm{isNonNegative}(q - p)$ and $p \leq q$ is decidable for every decimal fraction $p \in \mathbb{Z}[1/10]$ and $q \in \mathbb{Z}[1/10]$.
=--

+-- {: .num_prop}
###### Definition
**(Non-positive numbers)

Given $a \in \mathbb{Z}$ and $b \in \mathbb{N}$, a decimal fraction $a/10^b \in \mathbb{Z}[1/10]$ **is non-positive** if $a \leq 0$; i.e. the predicate $\mathrm{isNonPositive}(a/10^b)$ is defined as

$$\mathrm{isNonPositive}(a/10^b) \coloneqq a \leq 0$$
=--

+-- {: .num_prop}
###### Proposition
**(Decidability of non-positive numbers)

Given $a \in \mathbb{Z}$, $b \in \mathbb{N}$, the predicate $\mathrm{isNonPositive}(a/10^b)$ is a [[decidable proposition]].
=--

+-- {: .proof}
###### Proof

By definition, $\mathrm{isNonNegative}(a/10^b)$ is the same as $a \leq 0$. Because the relation $a \leq 0$ is decidable for every integer $a \in \mathbb{Z}$, $\mathrm{isNonPositive}(a/10^b)$ is decidable for every integer $a \in \mathbb{Z}$ and natural number $b \in \mathbb{N}$
=--

+-- {: .num_defn}
###### Definition
**(Greater than or equal to)

For every decimal fraction $p \in \mathbb{Z}[1/10]$ and $q \in \mathbb{Z}[1/10]$, there is a proposition $p \geq q$ called **greater than or equal to** and defined as

$$p \leq q \coloneqq \mathrm{isNonPositive}(q - p)$$
=--

+-- {: .num_prop}
###### Proposition
**(Decidability of greater than or equal to)

Given $p \in \mathbb{Z}[1/10]$ and $q \in \mathbb{Z}[1/10]$, the predicate $p \geq q$ is a [[decidable proposition]].
=--

+-- {: .proof}
###### Proof
By definition, $p \geq q$ is the same as $\mathrm{isNonPositive}(q - p)$. Because the relation $\mathrm{isNonPositive}(p)$ is decidable for every decimal fraction $p \in \mathbb{Z}[1/10]$, $\mathrm{isNonPositive}(q - p)$ and $p \geq q$ is decidable for every decimal fraction $p \in \mathbb{Z}[1/10]$ and $q \in \mathbb{Z}[1/10]$.
=--

+-- {: .num_prop}
###### Definition
**(Non-zero numbers)

Given $a \in \mathbb{Z}$ and $b \in \mathbb{N}$, a decimal fraction $a/10^b \in \mathbb{Z}[1/10]$ **is non-zero** if $a \neq 0$; i.e. the predicate $\mathrm{isNonZero}(a/10^b)$ is defined as

$$\mathrm{isNonZero}(a/10^b) \coloneqq a \neq 0$$
=--

+-- {: .num_prop}
###### Proposition
**(ordered integral domain)

The decimal fractions $(\mathbb{Z}[1/10], 0, +, -, 1, \cdot, \lt)$ form an [[ordered integral domain]].
=--

+-- {: .num_defn}
###### Definition
**(Ramp)

There exists a function $\mathrm{ramp}:\mathbb{Z}[1/10] \to \mathbb{Z}[1/10]$ called **ramp** and defined as

$$\mathrm{ramp}(a/10^b) \coloneqq \mathrm{ramp}(a)/10^b$$

for $a \in \mathbb{Z}$, $b \in \mathbb{N}$.
=--

+-- {: .num_defn}
###### Definition
**(Minimum)

There exists a binary operation $\min:\mathbb{Z}[1/10] \times \mathbb{Z}[1/10] \to \mathbb{Z}[1/10]$ called **minimum** defined as

$$\min(p,q) \coloneqq p - \mathrm{ramp}(p - q)$$

for $p \in \mathbb{Z}[1/10]$, $q \in \mathbb{Z}[1/10]$.
=--

+-- {: .num_defn}
###### Definition
**(Maximum)

There exists a binary operation $\max:\mathbb{Z}[1/10] \times \mathbb{Z}[1/10] \to \mathbb{Z}[1/10]$ called **maximum** defined as

$$max(p, q) \coloneqq p + \mathrm{ramp}(q - p)$$

for $p \in \mathbb{Z}[1/10]$, $q \in \mathbb{Z}[1/10]$.
=--

+-- {: .num_defn}
###### Definition
**(Absolute value)

There exists a function $\vert(-)\vert:\mathbb{Z}[1/10] \to \mathbb{Z}[1/10]$ called **absolute value** and defined as

$$\vert p \vert \coloneqq max(p, -p)$$

for $p \in \mathbb{Z}[1/10]$.
=--

+-- {: .num_prop}
###### Proposition
**(total order)

The decimal fractions $(\mathbb{Z}[1/10], 0, +, -, 1, \cdot, \lt, \gt, \leq, \geq, \min, \max, \mathrm{ramp})$ are a [[totally ordered ring]].
=--

### Scientific notation

+-- {: .num_prop}
###### Proposition
**(scientific notation)

For all decimal fractions $p \in \mathbb{Z}[1/10]$, if $(0 \lt p)$, then there exists an [[integer]] $b \in \mathbb{Z}$ such that $1 \leq a \cdot 10^b$ and $a \cdot 10^b \lt 10$.
=--

### Algebraic closure

The [[algebraic closure]] $\overline{\mathbb{Z}[1/10]}$ of the rational numbers is called the [[field]] of _[[algebraic numbers]]_, and is thus isomorphic to \overline{\mathbb{Q}}, the algebraic closure of the [[rational numbers]].

### Topologies

There are several interesting [[topological space|topologies]] on $\mathbb{Z}[1/10]$ that make $\mathbb{Z}[1/10]$ into a [[topological group]] under addition, allowing us to define interesting fields by taking the [[complete space|completion]] with respect to this topology:

1.  The _[[discrete topology]]_ is the most obvious, which is already complete.

2.  The _absolute-value topology_ is defined by the [[metric space|metric]] $d(x,y) \coloneqq {|x - y|}$; the completion is the field of [[real numbers]].

    (This topology is [[totally disconnected space|totally disconnected]].)

3.  For [[prime numbers]] $2$ and $5$, the _$2$-adic topology_ and _$5$-adic topology_ are defined by the [[ultrametric]] $d(x,y) \coloneqq 1/n$ where $n$ is the highest exponent on $2$ and $5$ respectively in the [[prime factorization]] of ${|x - y|}$; the completions of each metric are the fields of $2$-[[adic number|adic numbers]] and $5$-[[adic number|adic numbers]] respectively.

### Approximate division

Let $(-)\div(-):\mathbb{Z} \times \mathbb{Z}_{\neq 0} \to \mathbb{Z}$ be a [[Euclidean domain|Euclidean division function]] on the [[integers]]. Then there is a function called **approximate division**
$$(-)/(-): \mathbb{Z} \times \mathbb{Z}_{\neq 0} \to (\mathbb{N} \to \mathbb{Z}[1/10])$$
to a sequence of decimal fractions called the **infinite decimal representation** of the [[rational number]] $a/b$
$$(a/b)_n \coloneqq (a \cdot 10^n \div b)/10^n$$

This sequence in general does not [[converge]] in the decimal rational numbers; the [[initial object|initial]] [[archimedean integral domain]] in which every infinite decimal representation of a rational number converges is the [[rational numbers]].

## Related entries

* [[dyadic rational number]]

* [[decimal interval coalgebra]]

## References

* Wikipedia, _[Decimal](https://en.wikipedia.org/wiki/Decimal#Decimal_fractions)_

[[!redirects decimal rational numbers]]

[[!redirects decimal rational]]
[[!redirects decimal rationals]]

[[!redirects decimal fraction]]
[[!redirects decimal fractions]]


deck transformation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Generally, given a [[covering space]] $E \to X$, a *deck transformation* is a [[bundle]]-[[automorphism]], hence a [[continuous map]] $\delta \,\colon\, E \xrightarrow{\;\sim\;} E$ fixing the base $X$.

Assuming without real restriction that $X$ is [[connected topological space|connected]], then $E \to X$ is a [[principal bundle]] with [[discrete space|discrete]] [[fibers]] and the deck transformation determines and is determined by the element $\delta_x \,\colon\, E_x \xrightarrow{\;\sim\;} E_x$ of the [[permutation group]] $Aut(E_x)$ of its [[fiber]] $E_x$ over any fixed [[basepoint]] $x \in X$.

When the [[fundamental theorem of covering spaces]] applies (i.e. when $X$ is [[locally path-connected topological space|locally path-connected]] and [[semi-locally simply-connected topological space]]) then the [[fundamental group]] $\pi_1(X,x)$ has a canonical [[group action|action]] on this fiber, by a [[group homomorphism]] $\pi_1(X,x) \xrightarrow{\;} Aut(E_x)$, and often it is specifically this $\pi_1$-action which is referred to as being *by deck transformations*.

In particular, in the case that $E \to X$ is the [[universal cover]], then $E_x \,\simeq\, \pi_1(X)$ and $E \to X$ is a $\pi_1(X)$-[[principal bundle]] with the [[fundamental group]] $\pi_1(X)$ being identified with the group of deck transformations.

## Definition

+-- {: .num_defn}
###### Definition
A *deck transformation* or *cover automorphism* is an [[automorphism]] of a [[covering space]] relative to the base space.

i.e. if $p\colon E\to X$ is a cover then a cover automorphism $f\in deck(p)=\{f|f\in Aut(E), p\circ f=p\}\subseteq Aut(E)$ is an automorphism of $E$ such that $p$ is invariant under composition with $f$.
=--

## Related concepts

* [[branched cover]]

* [[twisted de Rham cohomology]]

* [[Borel-equivariant rational homotopy theory]]

## References

Any of the references at *[[covering space]]*.

See also:

* Wikipedia, *[Deck transformation](https://en.wikipedia.org/wiki/Covering_space#Deck_transformation)*

* MathWorld, *[Deck transformation](https://mathworld.wolfram.com/DeckTransformation.html)*

[[!redirects deck transformations]]

[[!redirects cover automorphism]]
[[!redirects cover automorphisms]]


Deck transformation > history

see *[[deck transformation]]*


decomposable differential form

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A [[differential form]] $\omega \in \Omega^n$ of homogeneous degree $n$ is said to be _decomposable_ if it is the [[wedge product]] of $n$ [[differential 1-form]] $(\alpha_k \in \Omega^1)_{k \in \{1, \cdots, n\}}$:

$$
  \omega = \alpha_1 \wedge \alpha_2 \wedge \cdots \wedge \alpha_n
$$

More generally in an $\mathbb{N}$-[[graded-commutative algebra]] an element of homogeneous degree may be called _decomposable_ if it may be written as the product of elements of degree 1.

## Related concepts

* [[decomposable tensor]]

* [[H-cohomology]]

## References

See also

* MathWorld, _[Decomposable](http://mathworld.wolfram.com/Decomposable.html)_

[[!redirects decomposable differential forms]]

[[!redirects decomposable form]]
[[!redirects decomposable forms]]


decomposable tensor

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A [[tensor]] -- an element in a [[tensor product]] of $k$ [[vector spaces]] $V_1\otimes V_2\otimes\ldots\otimes V_k$ -- is said to be __decomposable__ if it can be written in the form $v_1\otimes v_2\otimes\ldots\otimes v_k$ where $v_i \in V_i$ for $i = 1,\ldots, k$.

If all $V_i$ are copies of $V$ or $V^*$ for the same $V$ then we often talk of vectors in the tensor product as tensors and the tensor product the space of tensors. If $V_i$ has a basis $\{e^s_i\}_{s= 1}^{n_i}$ then $V_1\otimes V_2\otimes\ldots\otimes V_k$ has a basis consisting of all decomposable vectors of the form $e_1^{s_1}\otimes\ldots\otimes e_k^{s_k}$ for all $(s_1,\ldots,s_k)$ such that $1\leq s_i\leq n_i$.

Let $V$ be finite-dimensional vector space. Then for a tensor
$A\in V^{\otimes k}$ we say that $A$ has __decomposability rank__ $r$ if

$$
r = min\{ h | \exists A_1,\ldots, A_h decomposable, A = A_1+\ldots+A_h \}
$$

Distinguish this invariant from the covariance rank of a tensor.

While the decomposability rank of a covariance rank 2 tensor $A$ is the same as the rank of the corresponding matrix, for higher covariance rank tensors we do not have general algorithms how to determine the decomposability rank.

## Related concepts

* [[decomposable differential form]]

[[!redirects decomposable tensors]]

[[!redirects indecomposable tensor]]
[[!redirects indecomposable tensors]]


decomposition theorem for simplicial groups

#Discussion#

The following observation of Conduché is very useful when working with [[simplicial groupoid|simplicial groupoids]].

+--{.un_prop}
######Proposition
If $G$  is a simplicial group(oid), then $G_n$ decomposes as a multiple [[semidirect product]]:

$$G_n \cong NG_n \rtimes  s_0NG_{n-1}\rtimes  s_1NG_{n-1}\rtimes  s_1s_0NG_{n-2}
\rtimes  s_2NG_{n-1}\rtimes  \ldots s_{n-1}s_{n-2}\ldots s_0NG_0$$
=--

The order of the terms corresponds to a lexicographic ordering of the indices
$\emptyset$; 0; 1; 1,0; 2; 2,0; 2,1; 2,1,0; 3; 3,0; $\ldots$ and so on, the
term corresponding to $i_1 \gt \ldots \gt i_p$ being $s_{i_1}\ldots s_{i_p}NG_{n-p}$. The actions involved are clear once the following lemma is examined.

The proof  of the result is an induction based on a simple lemma, which is  easy to prove.

+--{.un_lemma}
######Lemma
If $G$ is a simplicial group(oid), then $G_n$ decomposes as a
semidirect product:
$$G_n \cong Ker d^n_n \rtimes  s^{n-1}_{n-1}(G_{n-1}).$$
=--

This decomposition generalises the one used in the classical [[Dold-Kan correspondence]]. It is extremely useful when analysing the [[Moore complex]] of a simplicial group and the relationship between that complex and the original simplicial group. It plays a crucial role in the theory of [[hypercrossed complexes]].

#Reference#
[[D. Conduché]], _Modules croisés généralisés de longueur 2_ , J. Pure Appl. Alg., 34, (1984), 155--178.

[[!redirects Conduché decomposition theorem]]


decomposition theory

Let $\mathcal{P}$ be a [[strictly full subcategory]] of a [[Grothendieck category]] $\mathcal{C}$ and $S$ a set. A __predecomposition theory__ is a function which to each $M$ in $\mathcal{P}$ gives a subset $\Gamma(M)\subset S$ such that

(i) $\Gamma(M) = 0$ iff $M = 0$

(ii) For any exact sequence

$$ 0\to M' \to M\to M'' \to 0 $$

in $\mathcal{C}$, with $M\in \mathcal{P}$,
$\Gamma(M')\subset \Gamma(M) \subset \Gamma(M')\cup\Gamma(M'')$.

(iii) If $M = \sum_i M_i$ then $\Gamma(M)=\cup_i \Gamma(M_i)$.

(iv) If $M'$ is an essential subobject of $M\in Ob(\mathcal{P})$, then $\Gamma(M') = \Gamma(M)$. One often writes, by abuse of notation, $\Gamma : \mathcal{P}\to S$.
For any $M\in\mathcal{P}$, the elements of $\Gamma(M)$ are called $\Gamma$-associates of $M$.

A class of examples are spectral predecomposition theories.

An object $M\in Ob(\mathcal{P})$ is said to be __$\Gamma$-coirreducible__
if $\Gamma(M)$ contains exactly one element.

A subobject $M'\subset M$ is __$\Gamma$-irreducible__ if
$M'/M$ is $\Gamma$-coirreducible.

A predecomposition theory $\Gamma:\mathcal{P}\to M$ is caleld a __decomposition theory__ if any subobject $M'$ of an object $M\in\mathcal{P}$ has a $\Gamma$-decomposition, that is the set of subobjects $\{M_i\}_{i\in I}$, $M_i\subset M$ such that

(D1) $\cap_{i\in I} M_i = M'$

(D2) for each $i$, $M_i$ is a $\Gamma$-irreducible subobject

(D3) $\Gamma(M/M_i) \cap \Gamma(M/M_j) = \emptyset$ if $i\neq j$

(D4) $\Gamma(M_i/M') = \Gamma(M/M')-\Gamma(M/M_i)$

(D5) $\Gamma(M/M') = \cup_i \Gamma(M/M_i)$

Examples include primary decomposition theory, Lesieur-Croisot [[tertiary decomposition theory]], Bourbaki's $\mathcal{P}$-primary decomposition theories...

* Nicolae Popescu, _An introduction to Abelian categories with applications to rings and modules_, Acad. Press 1973
* Joe W. Fisher, Harvey Wolff, _Decomposition theories for abelian categories_, Trans. Amer. Math. Soc. __182__ (1973), 61–69 [MR327870](http://www.ams.org/mathscinet-getitem?mr=327870) [doi](http://dx.doi.org/10.2307/1996520)

An earlier axiomatics in terms of pairs of objects is in

* John A. Riley, _Axiomatic primary and tertiary decomposition theory_, Trans. Amer. Math. Soc. 105 (1962), 177-201, [doi](http://dx.doi.org/10.1090/S0002-9947-1962-0141683-4 )
[[!redirects predecomposition theory]]
[[!redirects decomposition theories]]
[[!redirects predecomposition theories]]


decorated cospan

[[!redirects decorated cospans]]
# Decorated cospan

* table of contents
{: toc}

## Idea
Decorated cospans are a convenient formalism to deal with open networks, i.e. networks where some nodes are interpreted to be 'inputs' and some other to be 'outputs'. In fact, a natural way to do such labelling is to specify some function assigning the set of inputs to those nodes in the network tasked with receiving them, and analogously, a function assigning to outputs those nodes which provide them. This is, morally, a [[cospan]]: the network is the vertex, and inputs and outputs are the feet. However, the two things sit in 'different categories': e.g., inputs and outputs may be sets of wires and sockets, while the network has a more complicated description. Nevertheless, the network is just a set of nodes with more information attached, namely the way those nodes are connected. The key insight here is that to specify inputs/outputs of the network, we don't really care about this additional information. Hence we can use a 'cospan of nodes' (formally, a cospan in [[FinSet]]), and deal with the network structure later, as a _decoration_.

Surprisingly, this formalism naturally captures also other operations such as sequential and parallel composition of networks.

## Definition

### Categories of cospans
In a category $\mathbf C$ with finite colimits (or even just [[pushout|pushouts]]) [[cospans]] can be composed in a natural way. In fact, given $x \to s \leftarrow y$ and $y \to t \leftarrow z$, we get a new cospan $x \to p \leftarrow z$ by taking a pushout in the middle:
$$
  \array{
     &&&& s +_y t
     \\&
    &&
      {}^{p_s}\nearrow
      && \nwarrow^{p_t}
    \\
     && s &&&& t
      \\
      & {}^{f}\nearrow
      && \nwarrow^{g}
       &
       & {}^{h}\nearrow
      && \nwarrow^{i}
     \\
     x
     &&&&
     y
     &&&&
     z
  }
$$

Therefore cospans form a compositional structure: a category $\mathrm{Cospan}(\mathbf C)$ where objects are the same of $\mathbf C$ but morphisms from $x$ to $y$ are replaced by cospans with feet $x$ and $y$. Since the pushout we choose to form the composite of two cospans is, in general, unique only up to unique isomorphism, we actually need to define a $2$-category. Morphisms of cospans $(x \to p \leftarrow y) \Rightarrow (x \to q \leftarrow y)$ are given by any $\mathbf C$-morphism $\eta : p \to q$ making the following commute

$$
  \array{
    && p
    \\
    & {}^{f}\nearrow && \nwarrow^{g}
    \\
    x &&\downarrow^\eta&& z
    \\
    & {}_{f'}\searrow
    && \swarrow_{g'}
    \\
    && q
  }
$$

It is also customary to ignore the $2$-structure and simply work with $\mathrm{Cospan}(\mathbf C)$ as a $1$-category whose morphisms are _isomorphism classes_ of cospans.

### Categories of decorated cospans
+-- {: .un_defn #DecoratedCospanCategory}
###### Definition
Let $\mathbf C$ be a [[finitely cocomplete category|category with finite colimits]], and
$$
(F, \phi): (\mathbf C, +) \to (\mathbf {Set}, \times)
$$
be a [[lax monoidal functor]]. A **decorated cospan**, or more precisely an **$F$-decorated cospan**, is a pair $(x \overset{i}\to n \overset{o}\leftarrow y,\, {s}\in F n)$. We shall call the element $1 \overset{s}\to F n$ the **decoration** of the decorated cospan. A morphism of decorated cospans
$$
f : (x \overset{i}\to n \overset{o}\leftarrow y,\, s\in F n) \to (x \overset{i}\to n' \overset{o}\leftarrow y,\,  s'\in F n')
$$
is a morphism of cospans such that the following commutes:
$$
\array{
 && F n\\
 & \nearrow^s\\
 1 & & \downarrow^{F f}\\
 & \searrow^{s'}\\
 && F n'
}
$$
i.e. $F f(s)=F f(s')$.
=--

The definition is due to [Fong 2015](#BrendanFong2015), except there is allowed an arbitrary codomain, $(F, \phi): (\mathbf C, +) \to (\mathbf {D}, \otimes)$, and then a global element $I\overset{s}\to F n$. The notion is equivalent, by composing with the [[global elements]] functor $\mathbf{D}(I,-):\mathbf{D}\to \mathbf{Set}$, which is always lax monoidal. Nonetheless the $F$ in use will indeed typically arise from global elements of some familiar category $\mathbf{D}$.

+-- {: .un_def}
###### Definition
Given a cospan $x \to n \leftarrow y$ in $\mathbf C$, the **empty decoration on $n$** is the unique map
$$
1 \overset{\phi_1}\longrightarrow F\varnothing \overset{F!}\longrightarrow F n
$$
where $\varnothing$ is [[initial object|inital]] in $\mathbf C$ and $!$ denotes the universal morphism from such object.
=--

+-- {: .num_prop}
###### Proposition
([B. Fong 2015](#BrendanFong2015))
There is a category $F\mathrm{Cospan}$ of $F$-decorated cospans, with objects the objects of $\mathbf C$ and morphisms isomorphism classes of $F$-decorated cospans. Composition in this category is given by the class of the pushout of two representatives:
$$
  \array{
     &&&& n +_y m
     \\&
    &&
      {}^{j_n}\nearrow
      && \nwarrow^{j_m}
    \\
     && n &&&& m
      \\
      & {}^{i_x}\nearrow
      && \nwarrow^{o_y}
       &
       & {}^{i_y}\nearrow
      && \nwarrow^{o_z}
     \\
     x
     &&&&
     y
     &&&&
     z
  }
$$

along with the decoration

$$
1 \overset{(s,s')}\longrightarrow F n \times F m \overset{\phi_{n,m}}\longrightarrow F(n+m) \overset{F[j_n,j_m]}\longrightarrow F(n +_y m).
$$

where $1 \overset{s}\to F n$ and $1 \overset{s'}\to Fm$ are decorations of the first and second cospan, respectively.
=--
+-- {: .proof}
###### Proof
The identity morphism of an object $x$ in $F\mathrm{Cospan}$ is simply $x \overset{\mathrm{id}}\to x \overset{\mathrm{id}}\leftarrow x$ equipped with the empty decoration on $x$. The check that all relevant axioms are satisfied can be found in ([B. Fong 2015, Appendix A](#BrendanFong2015))
=--

+-- {: .un_remark}
###### Remark
The composition of decorations is the key construction of the decorated cospans formalisms. In fact, returning to the analogy with open networks, it constructs the composite network of a given 'link' of two networks. Hence the compositional structures of the cospans is leveraged to describe the compositional structure of a richer structure.
=--

+-- {: .un_remark}
###### Remark
Notice that the empty decoration gives a canonical way to decorate any cospan on $\mathbf C$. Indeed, it can be shown this defines a wide functor $\mathrm{Cospan}(\mathbf C) \embedsin F\mathrm{Cospan}$, as the composition of two empty-decorated cospans is again empty-decorated.
=--

### The monoidal and hypergraph structures
In the presence of a braiding on $\mathbf D$ (the 'decorating category'), the category of decorated cospans becomes not just [[symmetric monoidal category|symmetric monoidal]], but a full-blown [[hypergraph category]].

+-- {: .num_theorem}
###### Theorem
([B. Fong, 2015](#BrendanFong2015))
Let $\mathbf C$ be a category with finite colimts, $(\mathbf D, \otimes)$ a [[braided monoidal category]] and $(F, \phi) : (\mathbf C, +) \to (\mathbf D, \otimes)$ a lax braided monoidal functor. Then we may equip $F\mathrm{Cospan}$ with a symmetric monoidal and hypergraph structure, such that there is a [[wide subcategory|wide embedding]] of hypergraph categories
$$
\mathrm{Cospan}(\mathbf C) \embedsin F\mathrm{Cospan}.
$$
=--
+-- {: .proof}
###### Proof
We define the monoidal product of objects $x$ and $y$ of $F\mathrm{Cospan}$ to be their coproduct $x+y$ in $\mathbf C$, and defined the monoidal product of decorated cospans $(x \overset{i}\to n \overset{o}\leftarrow y, 1 \overset{s}\to F n)$ and $(x \overset{i}\to n' \overset{o}\leftarrow y, 1 \overset{s'}\to F n')$ to be
$$
  \array{
     && n + n'
      \\
      & {}^{i_x + i_{x'}}\nearrow
      && \nwarrow^{o_y + o_{y'}} &&,&& 1 \overset{\lambda^{-1}}\longrightarrow 1 \otimes 1 \overset{s \otimes s'}\longrightarrow F n \otimes F n' \overset{\phi_{n,n'}}\longrightarrow F(n+n')
     \\
     x+x'
     &&&&
     y+y'
  }
$$
The braiding in $\mathbf D$ can be now used to show this product is indeed functorial. Finally, we choose associator, unitors and braiding to be the images of those in $\mathrm{Cospan}(\mathbf C)$. The necessary checks are done in  ([B. Fong 2015, Appendix A](#BrendanFong2015)).

The hypergraph structure is defined by equipping each object $x \in F\mathrm{Cospan}$ with the image of the special commutative [[Frobenius algebra|Frobenius monoid]] specified by the hypergraph structure of $\mathrm{Cospan}(\mathbf C)$. The fact that the 'empty-decoration' embedding is an hypergraph functor is evident.
=--

If the monoidal unit of $(\mathbf D, \otimes)$ is the initial object, then each object admits only a decoration --- the empty one. This implies $\mathrm{Cospan}(\mathbf C)$ and $1_{\mathbf C}\mathrm{Cospan}$ are isomorphic as hypergraph categories.

## Examples

## Related concepts
* [[cospan]]
* [[structured cospan]]
* [[hypergraph category]]

## References ##

Decorated cospans were first defined by Brendan Fong:

* {#BrendanFong2015} [[Brendan Fong]], _Decorated Cospans_, ([arXiv:1502.00872](http://arxiv.org/abs/1502.00872)).


Dedekind completion

# Dedekind completions
* table of contents
{: toc}

## Idea

The _Dedekind completion_ of a [[linear order]] $L$ is a new strict linear order $\widebar{L}$ that contains [[suprema]] for all [[inhabited set|inhabited]] [[bounded subsets]], and such that a supremum in $L$ is still a supremum in $\widebar{L}$.

While Dedekind completeness was traditionally described in the context of the [[real numbers]], it can be stated for any [[strict linear order]], although it really works best for [[dense order|dense]] and unbounded (without [[top]] or [[bottom]]) strict linear orders. Intuitively, a strict linear order is _Dedekind complete_ if [[Dedekind cuts]] don't give any 'new' elements.

## Definitions

Let $S$ be a [[set]] equipped with the structure of a [[dense linear order]] without [[top]] or [[bottom]] elements (endpoints).

+-- {: .num_defn}
###### Definition

A __cut__ in $S$ is a pair of [[subsets]] $L, U \subset S$ of $S$ that satisfy the following eight properties:

1.  $L$ is [[inhabited subset|inhabited]];
2.  Dually, $U$ is inhabited;
3.  If $x \lt y \in L$, then $x \in L$;
4.  Dually, if $x \gt y \in U$, then $x \in U$;
5.  If $x \in L$, then $x \lt y \in L$ for some $y$;
6.  Dually, $x \in U$, then $x \gt y \in U$ for some $y$;
7.  If $x \lt y$, then $x \in L$ or $y \in U$;
8.  If $x \in L$ and $y \in U$, then $x \lt y$.
=--

+-- {: .num_defn}
###### Definition

The linearly ordered set $S$ is **Dedekind complete** if every cut $(L,U)$ is of the form
$$ L = \{x \;| \;x \lt a\},\; U = \{x \;|\; x \gt a\}$$
for some unique $a \in S$.
=--

+-- {: .num_defn}
###### Definition

If $T$ is also an unbounded dense strict linear order, $S$ is Dedekind complete, and we have a [[universal arrow]] $u\colon T \to S$ in the [[category]] of strict linear orders, then $S$ (equipped with $u$) is [[the]] **Dedekind completion** of $T$.
=--

The set of Dedekind cuts of [[rational numbers]] --the set of [[real numbers]]-- is Dedekind complete. In fact, starting with any unbounded dense linearly ordered set $S$, the set of Dedekind cuts is isomorphic to the reals as long as $S$ is a [[countably infinite set]].

The operation of forming the set of Dedekind cuts is idempotent, so the Dedekind completion can be constructed as the set of Dedekind cuts.  More precisely, the Dedekind-complete strict linear orders form a [[reflective subcategory]] of the category of dense unbounded strict linear orders, so that Dedekind completion is a kind of [[completion]] in the abstract categorial sense.

### Using open intervals ###

A set $F$ with a [[strict linear order]] is **Dedekind complete** if

* For all elements $a \in F$ and $b \in F$, the open interval $(a,b)$ is inhabited.

* For all elements $a \in F$, the upper unbounded open interval $(a,\infty)$ is inhabited.

* For all elements $a \in F$, the lower unbounded open interval $(-\infty,a)$ is inhabited.

* For all elements $a \in F$ and $b \in F$, $a \lt b$ if and only if $(b,\infty)$ is a subinterval of $(a,\infty)$

* For all elements $a \in F$ and $b \in F$, $b \lt a$ if and only if $(-\infty,b)$ is a subinterval of $(-\infty,a)$

* For all elements $a \in F$ and $b \in F$, if $a \lt b$, then $F$ is a subinterval of the union of $(a, \infty)$ and $(-\infty, b)$

* For all elements $a \in F$ and $b \in F$, the intersection of $(a,\infty)$ and $(-\infty,b)$ is a subinterval of the [[open interval]] $(a,b)$

## Generalisations

### In predicative constructive mathematics

In predicative constructive mathematics, subsets are large and therefore are proper classes. Thus, any Dedekind completion of a [[dense linear order]], and in particular the rational numbers, is a [[proper class]]. Instead, predicative constructive mathematicians typically use pairs of functions $L, U: \mathbb{Q} to \Sigma$ representing the open subsets of $\mathbb{Q}$ to define __predicative cuts__, where $\Sigma$ is a small [[sigma-frame|$\sigma$-frame]] with an [[embedding]] $\Sigma \to \mathrm{Prop}_\mathcal{U}$ into the large set of propositions for a universe $\mathcal{U}$. The definitions then become

+-- {: .num_defn}
###### Definition
Given a small [[sigma-frame|$\sigma$-frame]] $\Sigma$ that embeds into the large set of propositions $\mathrm{Prop}_\mathcal{U}$ for a universe $\mathcal{U}$ and a linearly ordered set $S$, a __cut__ in $S$ is a pair of functions $L, U: S to \Sigma$ that satisfy the following eight properties:

1.  There is an element $a \in S$ such that $L(a) = \top$.
2.  There is an element $b \in S$ such that $U(b) = \top$.
3.  If $a \lt b$ are elements of $S$ with $L(b) = \top$, then $L(a) = \top$.
4.  If $a \lt b$ are elements of $S$ with $U(a) = \top$, then $U(b) = \top$.
5.  If $L(a) = \top$, then there is an element of $S$ $a \lt b$ such that $L(b) = \top$.
6.  If $U(b) = \top$, then there is an element of $S$ $a \lt b$ for some $U(a) = \top$.
7.  If $a \lt b$ are elements of $S$, then $L(a) = \top$ or $U(b) = \top$.
8.  If $L(a) = \top$ and $U(b) = \top$, then $a \lt b$.
=--

+-- {: .num_defn}
###### Definition

The linearly ordered set $S$ is $\Sigma$-**Dedekind complete** if for every cut $(L,U)$ there exists a unique element $a \in S$ such that for all elements $x \in S$, $x \lt a$ implies $L(x) = \top$ and $x \gt a$ implies $U(x) = \top$.
=--

Given any linearly ordered set $S$, for any two $\sigma$-frames $\Sigma$ and $\Sigma^{'}$ that embed into $\mathrm{Prop}_\mathcal{U}$ such that $\Sigma \subseteq \Sigma^{'}$, if $A$ is the $\Sigma$-Dedekind completion of $S$ and $B$ is the $\Sigma^{'}$-Dedekind completion of $S$, then $A \subseteq B$.

### Strict preorders

+-- {: .standout}
Any paragraph containing the string 'duisp' is original research (although lower duisps at least are known in [[domain theory]]).
=--

One can generalise Dedekind completion from [[strict linear orders]] to [[strict preorders]].

+-- {: .num_defn}
###### Definition

A __duisp__ (**d**ense **u**nbounded **i**nhabited **s**trict **p**reorder) is a [[strictly preordered set]] $S$ such that, given finite (here always meaning [[Kuratowski-finite]]) [[subsets]] $F$ and $G$ of $S$ such that $x \lt z$ whenever $x \in F$ and $z \in G$, we have some $y$ in $S$ such that $x \lt y \lt z$ whenever $x \in F$ and $z \in G$.
=--

Note that, for $S$ a [[strict linear order]], $S$ is a duisp iff $S$ is dense, unbounded, and [[inhabited set|inhabited]], hence the term 'duisp'.  (Using linearity, we may assume that $F$ and $G$ are [[subsingletons]]; then two [[singleton subsets]] is denseness, one singleton subset and one [[empty subset]] is unboundedness, and two empty subsets is inhabitedness.)

+-- {: .num_defn}
###### Definition

Given a duisp $S$, a __cut__ is a pair $(L,U)$ of [[subsets]] such that:

1.  $L$ is [[inhabited subset|inhabited]] (which is a special case of (5) for $F$ the [[empty subset]]);
2.  Dually, $U$ is inhabited (a special case of (6));
3.  If $x \lt y \in L$, then $x \in L$;
4.  Dually, if $x \gt y \in U$, then $x \in U$;
5.  If $F$ is a finite subset of $L$ (which we may assume inhabited if we include (1)), then for some $x \in L$, every $y \in F$ satisfies $y \lt x$;
6.  Dually, if $F$ is a finite (inhabited) subset of $U$, then for some $x \in U$, every $y \in F$ satisfies $y \gt x$;
7.  If $L \lt x \lt U$ and $L \lt y \lt U$, then $x = y$;
8.  If $x \in L$ and $y \in U$, then $x \lt y$.
=--

We then define __Dedekind-complete__ duisps and __Dedekind completions__ of duisps the same as for dense linear orders, using this notion of cut.

A good example of a duisp is the set of [[rational number|rational]]-valued [[functions]] on any [[set]] $X$; the Dedekind completion is the set of [[real number|real]]-valued functions on $X$.

Sections 4.31--39 of _[[HAF]]_ do things in even more generality, but I don\'t really understand it yet.

## One-sided Dedekind completions

At least in [[classical mathematics]], considering only $L$ (for a __lower cut__) or $U$ (for an __upper cut__) doesn\'t really give us anything new for strict linear orders; we have only the technicality that $\infty \coloneqq (S,\empty)$ or $-\infty \coloneqq (\empty,S)$ is a cut (depending on the side), and we can rule even these out by simply requiring that $L$ have an upper bound or that $U$ have a lower bound.

In [[constructive mathematics]], one-sided cuts are more general; see [[one-sided real number]] for a discussion of the case where $S$ is the strict linear order of [[rational numbers]].

Even in classical mathematics, one-sided cuts do give us something new for strict preorders.  Here, we have first more general one-sided notions of duisps: a __lower duisp__ need only satisfy the condition of a duisp for $G$ a [[singleton]], and an __upper duisp__ need only satisfy the condition for $F$ a singleton.  Then the __lower Dedekind completion__ of a lower duisp is its set of lower cuts, and the __upper Dedekind completion__ of an upper duisp is its set of upper cuts.

For example, let $X$ be a [[compactum]] and let $S$ be the strictly preordered set of [[continuous map|continuous]] real-valued functions on $X$.  Then $S$ is a duisp, hence both a lower and upper duisp.  Its lower Dedekind completion is the set of lower [[semicontinuous map|semicontinuous]] functions on $X$ taking values in the [[lower reals]] (which classically are all either real or $\infty$); and dually on the upper side.  Even working classically and ignoring the technicality of $\infty$, semicontinuous functions are much more general than continuous ones.

## References

*  [[Paul Taylor]]\'s [page on Dedekind cuts](http://www.paultaylor.eu/ASD/dedras/classical)

*  Page 249&250 of _Continuous Lattices and Domains_ ([Google book](http://books.google.com/books?id=xPF9Hb7DPLgC)) covers lower Dedekind completions of duisps (although not under that name) in the context of [[domain theory]].

* Univalent Foundations Project, [[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]] (2013)

* Auke B. Booij, Extensional constructive real analysis via locators, ([arxiv:1805.06781](https://arxiv.org/abs/1805.06781))

* Steve Vickers, “Localic Completion Of Generalized Metric Spaces I”, [TAC](http://www.tac.mta.ca/tac/volumes/14/15/14-15abs.html)

* Davorin Lešnik, Synthetic Topology and Constructive Metric Spaces, ([arxiv:2104.10399](https://arxiv.org/abs/2104.10399))

[[!redirects Dedekind completion]]
[[!redirects Dedekind completions]]
[[!redirects Dedeking completion]]

[[!redirects Dedekind-complete]]
[[!redirects Dedekind complete]]
[[!redirects Dedekind completeness]]

[[!redirects Dedekind-complete order]]
[[!redirects Dedekind-complete orders]]
[[!redirects Dedekind complete order]]
[[!redirects Dedekind complete orders]]

[[!redirects sigma Dedekind completion]]
[[!redirects sigma Dedekind completions]]
[[!redirects sigma Dedeking completion]]
[[!redirects sigma-Dedekind completion]]
[[!redirects sigma-Dedekind completions]]
[[!redirects sigma-Dedeking completion]]
[[!redirects Sigma Dedekind completion]]
[[!redirects Sigma Dedekind completions]]
[[!redirects Sigma Dedeking completion]]
[[!redirects Sigma-Dedekind completion]]
[[!redirects Sigma-Dedekind completions]]
[[!redirects Sigma-Dedeking completion]]

[[!redirects sigma Dedekind-complete]]
[[!redirects sigma Dedekind complete]]
[[!redirects sigma Dedekind completeness]]
[[!redirects sigma-Dedekind-complete]]
[[!redirects sigma-Dedekind complete]]
[[!redirects sigma-Dedekind completeness]]
[[!redirects Sigma Dedekind-complete]]
[[!redirects Sigma Dedekind complete]]
[[!redirects Sigma Dedekind completeness]]
[[!redirects Sigma-Dedekind-complete]]
[[!redirects Sigma-Dedekind complete]]
[[!redirects Sigma-Dedekind completeness]]

[[!redirects sigma Dedekind-complete order]]
[[!redirects sigma Dedekind-complete orders]]
[[!redirects sigma Dedekind complete order]]
[[!redirects sigma Dedekind complete orders]]
[[!redirects sigma-Dedekind-complete order]]
[[!redirects sigma-Dedekind-complete orders]]
[[!redirects sigma-Dedekind complete order]]
[[!redirects sigma-Dedekind complete orders]]
[[!redirects Sigma Dedekind-complete order]]
[[!redirects Sigma Dedekind-complete orders]]
[[!redirects Sigma Dedekind complete order]]
[[!redirects Sigma Dedekind complete orders]]
[[!redirects Sigma-Dedekind-complete order]]
[[!redirects Sigma-Dedekind-complete orders]]
[[!redirects Sigma-Dedekind complete order]]
[[!redirects Sigma-Dedekind complete orders]]


Dedekind cut

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
=--
=--

# Dedekind cuts
* table of contents
{: toc}

## Idea

What came to be called *Dedekind cuts* (a notion due to [Dedekind (1872)](#Dedekind1872)) is a way to make precise the idea that a [[real number]] is that which can be approximated (in the [[absolute value]] metric) by [[rational numbers]].

[Dedekind 1872](#Dedekind1872) pointed out that a real number may be uniquely determined its order relationships with rational numbers.  That is, the real number $x$ is determined by its __lower set__ $L_x$ and its __upper set__ $U_x$:
\[ \label{cuts} \begin {split}
   L_x \coloneqq \{ a\colon \mathbb{Q} \;|\; a \lt x \} ,\\
   U_x \coloneqq \{ b\colon \mathbb{Q} \;|\; x \lt b \}
.\end {split} \]
Dedekind had found great success understanding the 'ideal numbers' of [[ring theory]] as certain sets, which are what we now understand as [[ideal of a ring|ideals]].  So he defined a 'real number' as a pair of sets of rational numbers, the lower and upper sets shown above (actually a slight variation).  Such a pair of sets of rational numbers he called a 'Schnitt' (English 'cut'), nowadays called a 'Dedekind cut'.

## Definitions {#Definitions}

Exactly which pairs of sets of rational numbers can appear this way?  We will define a __Dedekind cut__ to be any pair $(L,U)$ of sets of rational numbers satisfying these conditions:

1.  $L$ is [[inhabited set|inhabited]]; that is, some rational number belongs to $L$.
2.  Similarly, $U$ is inhabited.
3.  $L$ is a [[lower set]]; that is, if $a \lt b$ are rational numbers with $b \in L$, then $a \in L$.
4.  Similarly, $U$ is an [[upper set]]: if $a \lt b$ are rational numbers with $a \in U$, then $b \in U$.
5.  $L$ is an upwards [[open set]]; that is, if $a \in L$, then $a \lt b$ for some $b \in L$.
6.  Similarly, $U$ is a  downwards [[open set]]: if $b \in U$, then $a \lt b$ for some $a \in U$.
7.  If $a \lt b$ are rational numbers, then $a \in L$ or $b \in U$.
8.  If $a \in L$ and $b \in U$, then $a \lt b$.

Actually, this definition is redundant; any pair $(L,U)$ that satisfies (7&8) must also satisfy (3&4).  However, we state (3&4) explicitly to simplify the description of the variations below.

If $x \coloneqq (L,U)$ is a Dedekind cut, then we define $a \lt x$ to mean that $a \in L$ and define $x \lt b$ to mean that $b \in U$.

### Using open intervals

This definition is due to [Vickers15](#Vickers15).

Let us define the family of upwards unbounded [[open intervals]] $\left(]a,\infty[\right)_{a \in \mathbb{Q}}$ and lower unbounded [[open intervals]] $\left(]-\infty, a[\right)_{a \in \mathbb{Q}}$. The set of **Dedekind cuts** is a [[frame]] generated by $]a,\infty[$ and $]-\infty, a[$ such that

$$\mathbb{Q} \subseteq \bigcup_{a \in \mathbb{Q}} ]a,\infty[$$

$$\mathbb{Q} \subseteq \bigcup_{a \in \mathbb{Q}} ]-\infty,a[$$

* for all $a \in \mathbb{Q}$ and $b \in \mathbb{Q}$,
$$(a \lt b) \implies (]b,\infty[ \subseteq ]a,\infty[)$$

* for all $a \in \mathbb{Q}$ and $b \in \mathbb{Q}$,
$$(b \lt a) \implies (]-\infty,b[ \subseteq ]-\infty,a[)$$

* for all $a \in \mathbb{Q}$
$$]a,\infty[ \subseteq \bigcup_{b \in ]a,\infty[} ]b,\infty[$$

* for all $a \in \mathbb{Q}$
$$]-\infty,a[ \subseteq \bigcup_{b \in ]-\infty,a[} ]-\infty,b[$$

* for all $a \in \mathbb{Q}$ and $b \in \mathbb{Q}$,
$$]a \lt b[ \Rightarrow \mathbb{Q} \subseteq ]a, \infty[ \cup ]-\infty, b[$$

* for all $a \in \mathbb{Q}$ and $b \in \mathbb{Q}$,
$$]a,\infty[ \cap ]-\infty,b[ \subseteq ]a,b[$$

A **Dedekind cut** is an element of this frame.

### Structural Dedekind cuts

There is a structural version of Dedekind cuts in [[structural set theory]]. A *structural subset* of the rational numbers is a set $A$ with an [[injection]] $i_A:A \hookrightarrow \mathbb{Q}$. We say that a rational number $q$ belongs to $A$ if there exists an element $x \in A$ such that $i_A(x) = q$.
$$\exists x \in A.i_A(x) = q$$

A **structural Dedekind cut** consists of two structural subsets $(L, i_L)$ and $(U, i_U)$ with [[injections]] $i_L:L \hookrightarrow \mathbb{Q}$ and $i_R:R \hookrightarrow \mathbb{Q}$ such that

1.  $L$ is [[inhabited set|inhabited]]; that is, some rational number belongs to $L$.
2.  Similarly, $U$ is inhabited.
3.  $L$ is a [[lower set]]; that is, if $a \lt b$ are rational numbers such that $b$ belongs to $L$, then $a$ belongs to $L$.
4.  Similarly, $U$ is an [[upper set]]: if $a \lt b$ are rational numbers with $a$ belongs to $U$, then $b$ belongs to $L$.
5.  $L$ is an upwards [[open set]]; that is, if $a$ belongs to $L$, then $a \lt b$ for some $b$ that belongs to $L$.
6.  Similarly, $U$ is a  downwards [[open set]]: if $b$ belongs to $U$, then $a \lt b$ for some $a$ that belongs to $U$.
7.  If $a \lt b$ are rational numbers, then $a$ belongs to $L$ or $b$ belongs to $U$.
8.  If $a$ belongs to $L$ and $a$ belongs to $U$, then $a \lt b$.

## Motivation

The point of condition (7) is that we can estimate $x$ as closely as we like by choosing appropriate rational numbers.

This is entirely [[constructive mathematics|constructive]].  We have $a_0 \lt x$ for some $a_0$ (by 1) and $x \lt b_0$ for some $b_0$ (by 2), so if we wish to estimate $x$ to within a positive rational number $\epsilon$, then we simply let $n$ be $\lceil{2/\epsilon}\rceil$ and apply (7) to each of the finitely many ($\lceil{n(b_0-a_0)}\rceil$) rational numbers with denominator $n$ that lie between $a_0$ and $b_0$.  This will eventually give us a numerator $i$ such that $(i-1)/n \lt x \lt (i+1)/n$, so we have estimated $x$ within $2/n \leq \epsilon$.  Alternatively, to estimate $x$ to $n$ digits after a decimal point (with an uncertainty of $1$ in the last digit), apply (7) to each of the $\lceil{(b_0-a_0) \times 10^n}\rceil$ numbers between $a_0$ and $b_0$ that have $n$ digits after the decimal point.  (We can never constructively determine the final digit with perfect certainty; for example, if we successively estimate $x$ as $0.5, 0.50, 0.500, \ldots$, no finite number of such estimates can ever ensure that $x$ won\'t come out to $0.4999\cdots9998$ when we go one digit farther.)

Besides giving us a place to begin our estimation, (1&2) ensure that $x$ is finite.  Condition (8) enforces the [[transitive relation|transitivity]] law $a \lt x \lt b \implies a \lt b$; (3&4) are likewise forms of transitivity.  (5&6) ensure that, even if $x$ happens to be a rational number, we are using the sets $L_x$ and $U_x$ given in (eq:cuts) instead of their closures (defined with $\leq$ instead of $\lt$).

## Properties

We define $x \lt y$ to mean that there exists a rational number $a$ such that $x \lt a \lt y$; that is, some rational number belongs to both the upper set of $x$ and the lower set of $y$.  It is then straightforward to prove that $\lt$ is a [[linear order]]:

*  [[asymmetric relation|Asymmetry]]:  If $x \lt y$ and $y \lt x$, pick $a$ such that $x \lt a \lt y$ and $b$ such that $y \lt b \lt x$.  Applying (8) in two ways, $a \lt b$ and $b \lt a$, which is impossible.
*  [[comparison relation|Comparison]]:  If $x \lt z$ and $y$ is a Dedekind cut, then pick $a$ such that $x \lt a \lt z$.  Using (5) or (6), we can generalise this to two rational numbers $a$ and $b$ such that $x \lt a \lt b \lt z$.  Applying (7) to $a \lt b$, we get $x \lt y$ (if $a \lt y$) or $y \lt z$ (if $y \lt b$).
*  [[connected relation|Connectedness]]:  Suppose that $x \lt y$ and $y \lt x$ both fail, and suppose that $a \lt y$.  By (5), we have $a \lt b \lt y$ for some $b$.  Applying (7), we have $a \lt x$ or $x \lt b$.  Since $x \lt b \lt y$ contradicts our assumptions, we have $a \lt x$.  In summary, $a \lt y \implies a \lt x$.  We can prove $a \lt x \implies a \lt y$, $b \lt x \implies b \lt y$, and $b \lt y \implies b \lt x$ similarly.  Therefore, $x$ and $y$ are equal (as pairs of subsets).

As usual with a linear order, we define $x \leq y$ to mean that $y \lt x$ is false.  In particular, connectedness of $\lt$ corresponds to [[antisymmetric relation|antisymmetry]] of $\leq$.

We can also derive a Dedekind cut from any rational number $x$ by taking (eq:cuts) to be the definition of $x$ as a Dedekind cut.  Thus every rational number is interpreted as a real number.  This is a strict inclusion of linear orders:

*  If $x \lt y$ as rational numbers, then let $a$ be the rational number $(x+y)/2$; we have $x \lt a \lt y$, so $x \lt y$ as real numbers.

In this way, the Dedekind cuts form an extension of the linearly ordered set of rational numbers.  More than that, this extension is [[Dedekind completion|Dedekind-complete]], which basically means that you cannot extend further by taking Dedekind cuts of Dedekind cuts.

To be explicit, suppose that $(L,U)$ is a pair of sets of Dedekind cuts (rather than of rational numbers) that satisfies (1--8), using Dedekind cuts in place of rational numbers.  Let $\sup L$ be the union of the left halves of the cuts that appear in $L$, and let $\inf U$ be the union of the right halves of the cuts that appear in $U$.  Then one can prove that $x \coloneqq (\sup L, \inf U)$ is a Dedekind cut (of rational numbers), and furthermore that $L$ and $U$ may be recovered from $x$ as in (eq:cuts), again using Dedekind cuts in place of rational numbers.

We say that the Dedekind cuts form the [[Dedekind completion]] of the linear order $\mathbb{Q}$.

+-- {: .num_remark}
###### Remark

Being a member of the lower (resp. the upper) set of a Dedekind cut is almost [[stable proposition|stable]] (under [[double negation]]): For any rational number $a$ it holds that $a \in L$ iff there exists a rational number $b \gt a$ such that $\neg\neg(b \in L)$. "Only if" is by condition (5). "If" is by condition (7): We have $a \in L$ or $b \in U$. In the first case we're done; in the second case it follows that $\neg(b \in L)$ by (8), which is a contradiction to $\neg\neg(b \in L)$.
=--

+-- {: .num_remark}
###### Remark

Equality of Dedekind cuts is [[stable proposition|stable]] (under [[double negation]]).
=--

## Variations

A Dedekind cut is, in full clarity, a __bounded, open, rounded, located, two-sided Dedekind cut of rational numbers__.  But there are several simple variations on the definition above, many of which may be found in the literature.

### Overdetermined cuts

The conditions (5&6) are not really necessary.  If you leave them out, then the result that $\lt$ is a [[connected relation]] on Dedekind cuts fails; but instead we identify cuts $x$ and $y$ whenever $x \leq y \leq x$.  Then the linearly ordered set of real numbers becomes a [[quotient set]] of the set of cuts.  In a similar vein, we can weaken (3,4,7) as follows:

* 3′:  If $a \lt b$ are rational numbers with $b \in L$, then $a' \in L$ for some $a' \lt a$.
* 4′:  If $a \lt b$ are rational numbers with $a \in U$, then $b' \in U$ for some $b' \gt b$.
* 7′:  If $a \lt b$ are rational numbers, then $a' \in L$ for some $a' \lt a$ or $b' \in U$ for some $b' \gt b$.

Just as (3,4) follow from (7,8), so (3′,4′) follow from (7′,8).  Again, $\lt$ is no longer connected, but we still get a linear order on a quotient set.  The weakest axiom system that will give the correct linear order on the quotient is (1,2,7′,8).  Incidentally, you can also recover the original definition using (3,4,7′) in place of (7).

To emphasise that one does require (5,6), one may speak of an __open cut__.  To emphasise that one does require (3,4), one may speak of a __rounded cut__.

Note that we may elegantly (although requiring a fancier [[logic]]) combine (3,4,5,6) into two conditions as follows:

* (3&5):  A rational number $a$ belongs to $L$ if and only if there exists a rational number $b \in L$ such that $a \lt b$.
* (4&6):  A rational number $b$ belongs to $U$ if and only if there exists a rational number $a \in U$ such that $a \lt b$.

### Covering cuts

Another approach is to *strengthen* (7):

* 7″:  Every rational number belongs to $L$ or $U$.

While (7″) alone does not imply (7), it does together with (8).  Also note that, using (8), we can even change 'or' to '[[xor]]' in (7″).

The cuts that satisfy (1--6, 7″, 8) are precisely the cuts that define [[irrational number]]s.  If we remove either (5) or (6), then the resulting axioms define one cut for each rational number as well.  If we remove *both* (5) and (6), then we get two cuts for each rational number and must (as in the previous section) pass to a quotient; this was actually Dedekind\'s original 1872 system.  These versions are not acceptable [[constructive mathematics|constructively]] (and we cannot even prove that they are closed under addition), but they work fine if you use [[excluded middle]].  (Actually, the somewhat weaker [[limited principle of omniscience]] seems to suffice.)

### One-sided cuts

Suppose that we consider only $L$ and demand (1,3,5), the axioms that refer only to $L$.  Any such set $L$ of rational numbers is a __lower cut__.  Similarly, a set $U$ that satisfies (2,4,6) is an __upper cut__.  If we demand that the set $L$ or $U$ is a [[proper subset]], then we have a __bounded__ lower/upper cut.  If we wish to emphasise that we are discussing a cut as defined above, with both $L$ and $U$, we speak of a __two-sided cut__.

Assuming [[excluded middle]], every bounded lower or upper cut generates a two-sided cut, by whichever of these rules is needed:
\[ \label{twosiding} \begin {split}
   U \coloneqq \{ b\colon \mathbb{Q} \;|\; \forall a \in L,\; a \lt b \} ,\\
   L \coloneqq \{ a\colon \mathbb{Q} \;|\; \forall b \in U,\; a \lt b \}
.\end {split} \]
In addition, the unbounded lower cut $L \coloneqq \mathbb{Q}$ represents $\infty$, while the unbounded upper cut $U \coloneqq \mathbb{Q}$ represents $-\infty$.

In [[constructive mathematics]], one-sided cuts define a potentially more general notion of real number, called a [[lower real]] or [[upper real]] (with the same direction as the cut).  Unlike the covering cuts in the previous section, these are actually of interest constructively.  If we wish to emphasise or clarify that we are discussing a two-sided cut as in our original definition, rather than a one-sided cut which has been made two-sided through (eq:twosiding) (which may not satisfy (7) constructively), then we may speak of a __located cut__.

### Unbounded cuts

If we drop conditions (1&2), then the result is an __extended cut__; these represent [[extended real number]]s.  If we wish to emphasise or clarify that we are discussing a cut that satisfies (1,2), then we may speak of a __bounded cut__.

By [[excluded middle]], every extended cut is either a Dedekind cut, $(\mathbb{Q},\empty)$, or $(\empty,\mathbb{Q})$.  While the bounded cuts represent real numbers, the other two extended cuts represent $\pm\infty$.

### Interval cuts
{#intervals}

If we drop (7) (but keep 3&4), then the result is an __[[interval cut]]__; these represent intervals of the form $[x,y]$ for $x \leq y$, where $x$ is a lower real and $y$ is an upper real.  If we use only (1--6), then we allow for __back-to-front intervals__ $[x,y]$ where $x \leq y$ is no longer required.  Even classically, we cannot represent back-to-front intervals as subsets of the real line, but it is still possible to compute with them.

Whereas a located Dedekind cut can be approximated as closely as we wish, an interval cut represents a number as we measure it in the real world, where we may not be technically able (or even theoretically able, as in [[quantum physics]]) to make a more precise measurement.  A back-to-front interval indicates that we have made contradictory measurements, perhaps as a result of experimental error.

An article on [[interval arithmetic]] is probably the right place to talk about these systems (although there are many varieties of interval arithmetic).

### Predicative cuts

In predicative mathematics, subsets are large and therefore are proper classes. Instead, in the manner of [[formal topology]], they typically use pairs of functions $L, U: \mathbb{Q} to \Sigma$ representing the open subsets of $\mathbb{Q}$ to define __$\Sigma$-Dedekind cuts__, where $\Sigma$ is a small $\sigma$-locale, or small [[sigma-frame|$\sigma$-frame]] with an [[embedding]] $\Sigma \to \mathrm{Prop}_\mathcal{U}$ into the large set of propositions for a universe $\mathcal{U}$. The definitions then become

1.  There is a rational number $a$ such that $L(a) = \top$.
2.  There is a rational number $b$ such that $U(b) = \top$.
3.  If $a \lt b$ are rational numbers with $L(b) = \top$, then $L(a) = \top$.
4.  If $a \lt b$ are rational numbers with $U(a) = \top$, then $U(b) = \top$.
5.  If $L(a) = \top$, then there is a rational number $a \lt b$ such that $L(b) = \top$.
6.  If $U(b) = \top$, then there is a rational number $a \lt b$ for some $U(a) = \top$.
7.  If $a \lt b$ are rational numbers, then $L(a) = \top$ or $U(b) = \top$.
8.  If $L(a) = \top$ and $U(b) = \top$, then $a \lt b$.

For any two $\sigma$-frames $\Sigma$ and $\Sigma^{'}$ that embed into $\mathrm{Prop}_\mathcal{U}$ such that $\Sigma \subseteq \Sigma^{'}$, if $A$ is the set of  $\Sigma$-Dedekind cuts of $Q$ and $B$ is the $\Sigma^{'}$-Dedekind cuts of $S$, then $A \subseteq B$.

### Cuts of other numbers

Instead of starting with rational numbers, we could use any [[dense suborder|dense]] subset of $\mathbb{Q}$; the result would be equivalent.  For example, we could start with the [[dyadic number]]s to show more explicitly how real numbers are represented by rational numbers written in binary notation.  Alternatively, we could start with something larger than (or incomparable with) $\mathbb{Q}$ that is still a dense subset of $\mathbb{R}$ (although you have to know what $\mathbb{R}$ is to state this).  For example, we could start with the real [[algebraic number]]s or the [[computable number]]s.

The basic theory also generalises immediately to any unbounded, [[dense linear order]] $S$; again, the result is equivalent as long as $S$ is a [[countably infinite set]].  The theory may be generalised further even to orders which may not be unbounded, dense, or even linear, but this requires modifications; see [[Dedekind completion]] for unbounded dense [[quasiorders]] or unbounded dense [[irreflexive comparisons]] or Sections 4.31--39 of _[[HAF]]_ for even more generality.

## See also

* [[locator]]

* [[Dedekind cut structure]]

* [[streak]]

## References

The original article:

* {#Dedekind1872} [[Richard Dedekind]], _Stetigkeit und irrationale Zahlen_ (1872) [[Google Books](http://books.google.com/books?id=AO4GAAAAYAAJ), retyped: [web](http://www.math.ru.nl/werkgroepen/gmfw/bronnen/dedekind2.html)]

* [[Richard Dedekind]] (transl. by W. Beman), _Continuity and irrational numbers_, Chapter I in: *Essays on the Theory of Numbers*, Chicago (1901) [[Project Gutenberg](http://www.gutenberg.org/ebooks/21016), [pdf](https://www.gutenberg.org/files/21016/21016-pdf.pdf)]

Review:

*  [[Eric Schechter]], 4.31-4.39 in *[[Handbook of Analysis and its Foundations]]*, Academic Press (1996) [[doi:10.1016/B978-0-12-622760-4.X5000-6](https://doi.org/10.1016/B978-0-12-622760-4.X5000-6)]

Discussion in [[locale]]-theory:

* {#Vickers15} [[Steve Vickers]], *Localic Completion Of Generalized Metric Spaces I*, Theory and Applications of Categories **14** 15 (2005) 328-356 [[tac:14-15](http://www.tac.mta.ca/tac/volumes/14/15/14-15abs.html)]

Discussion via [[abstract Stone duality]]:

*  [[Paul Taylor]], *[Dedekind cuts](http://www.paultaylor.eu/ASD/dedras/classical)* (2007-2009?)

* [[Andrej Bauer]], [[Paul Taylor]], *The Dedekind reals in abstract Stone duality*, Mathematical Structures in Computer Science **19** 4 (2009) 757-838 [[doi:10.1017/S0960129509007695](https://doi.org/10.1017/S0960129509007695)]

review in:

* [[Andrej Bauer]], *Efficient Computation with Dedekind Reals*, talk at *[Computability and complexity in analysis 2008](https://math.andrej.com/2008/08/24/efficient-computation-with-dedekind-reals/)* and at *[Mathematics, Algorithms and Proofs 2008](http://cdsagenda5.ictp.trieste.it/full_display.php?smr=0&ida=a07167)* (2008) [[web](https://math.andrej.com/2008/08/24/efficient-computation-with-dedekind-reals/), slides: [pdf](https://math.andrej.com/wp-content/uploads/2008/08/slides-map2008.pdf), extended abstract: [pdf](https://math.andrej.com/wp-content/uploads/2008/08/abstract-cca2008.pdf)]

Discussion in [[synthetic topology]]:

* Davorin Lešnik, Synthetic Topology and Constructive Metric Spaces [[arxiv:2104.10399](https://arxiv.org/abs/2104.10399)]

Formalization in [[homotopy type theory]]:

* {#UFP13} [[Univalent Foundations Project]], §11.2 *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

* [[Auke B. Booij]], Extensional constructive real analysis via locators, ([abs:1805.06781](https://arxiv.org/abs/1805.06781))

[[!redirects Dedekind cut]]
[[!redirects Dedekind cuts]]
[[!redirects Dedekind real]]
[[!redirects Dedekind reals]]
[[!redirects Dedekind real number]]
[[!redirects Dedekind real numbers]]

[[!redirects material Dedekind cut]]
[[!redirects material Dedekind cuts]]
[[!redirects material Dedekind real]]
[[!redirects material Dedekind reals]]
[[!redirects material Dedekind real number]]
[[!redirects material Dedekind real numbers]]

[[!redirects structural Dedekind cut]]
[[!redirects structural Dedekind cuts]]
[[!redirects structural Dedekind real]]
[[!redirects structural Dedekind reals]]
[[!redirects structural Dedekind real number]]
[[!redirects structural Dedekind real numbers]]

[[!redirects two-sided Dedekind cut]]
[[!redirects two-sided Dedekind cuts]]

[[!redirects sigma Dedekind cut]]
[[!redirects sigma Dedekind cuts]]
[[!redirects sigma-Dedekind cut]]
[[!redirects sigma-Dedekind cuts]]
[[!redirects Sigma Dedekind cut]]
[[!redirects Sigma Dedekind cuts]]
[[!redirects Sigma-Dedekind cut]]
[[!redirects Sigma-Dedekind cuts]]

[[!redirects sigma Dedekind real]]
[[!redirects sigma Dedekind reals]]
[[!redirects sigma-Dedekind real]]
[[!redirects sigma-Dedekind reals]]
[[!redirects Sigma Dedekind real]]
[[!redirects Sigma Dedekind reals]]
[[!redirects Sigma-Dedekind real]]
[[!redirects Sigma-Dedekind reals]]

[[!redirects sigma Dedekind real number]]
[[!redirects sigma Dedekind real numbers]]
[[!redirects sigma-Dedekind real number]]
[[!redirects sigma-Dedekind real numbers]]
[[!redirects Sigma Dedekind real number]]
[[!redirects Sigma Dedekind real numbers]]
[[!redirects Sigma-Dedekind real number]]
[[!redirects Sigma-Dedekind real numbers]]


Dedekind cut structure

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Definition

Given a set $T$ with a [[dense linear order]] $\lt$, a pair of [[subsets]] $(L, R)$ of $T$ with [[injections]] $i_L:L \to T$ and $i_R:R \to T$ is a **Dedekind cut structure** if it comes equipped with the following structure

* an [[element]] $l \in L$
* an element $r \in R$
* for every element $a \in L$ and $b \in T$, a [[function]]
$$c_d(a, b):]b, a[ \to \{c \in L \vert i_L(c) = b\}$$
* for every element $a \in R$ and $b \in T$, a function
$$c_u(a, b):]a, b[ \to \{c \in R \vert i_R(c) = b\}$$
* for every element $a \in L$, an element
$$o_d(a) \in \{b \in L \vert i_L(a) \lt i_L(b)\}$$
* for every element $a \in R$, an element
$$o_u(a) \in \{b \in R \vert i_R(b) \lt i_R(a)\}$$
* for every element $a \in L$ and $b \in R$, an element
$$t(a, b) \in ]i_L(a), i_R(b)[$$
* for every element $a \in T$ and $b \in T$, a function
$$L(a, b):]a,b[ \to (\{c \in L \vert i_L(c) = a\} \uplus \{c \in R \vert i_R(c) = b\})$$

where $]a, b[$ is the [[open interval]] bounded by $a$ from below and by $b$ from above.

## See also ##

* [[Dedekind cut]]
* [[locator]]
* [[dense linear order]]

## References

* [[Auke Booij]], *Extensional constructive real analysis via locators*, ([abs:1805.06781](https://arxiv.org/abs/1805.06781))

[[!redirects Dedekind cut structures]]


Dedekind domain

# Contents
* table of contents
{: toc}

## Definition

In [[algebra]], a __Dedekind domain__ is an [[integral domain]] in which every non-[[zero ideal|zero]] [[proper ideal]] factors into a [[product of ideals|product]] of [[prime ideals]].

## Examples

* The [[ring of integers]] in any [[number field]] is a Dedekind domain.

## Related concepts

* [[integral domain]]

## References

* Wikipedia, _[Dedekind domain](http://en.wikipedia.org/wiki/Dedekind_domain)_

[[!redirects Dedekind domain]]
[[!redirects Dedekind domains]]


Dedekind eta function

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Theta functions
+--{: .hide}
[[!include theta functions - contents]]
=--
#### Elliptic cohomology
+-- {: .hide}
[[!include elliptic cohomology -- contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[modular form]].

The [[Jacobi theta function]] for special values of its arguments...

## Definition

$$
  \eta(\tau) = q^{1/24} \prod_{n=1}^\infty (1-q^n)
$$

for $q \coloneqq e^{2\pi i \tau}$.

## Properties

### As functional determinant of Laplace operator on elliptic curve

For $\mathbb{C}/(\mathbb{Z}\oplus \tau \mathbb{Z})$
a [[complex torus]] (complex [[elliptic curve]]) equipped with its
standard flat [[Riemannian metric]], then the [[zeta function of an elliptic differential operator|zeta function]] of the corresponding [[Laplace operator]] $\Delta$ is

$$
  \zeta_{\Delta} = (2\pi)^{-2 s} E(s)
   \coloneqq
   (2\pi)^{-2 s} \underset{(k,l)\in \mathbb{Z}\times\mathbb{Z}-(0,0)}{\sum}
  \frac{1}{{\vert k +\tau l\vert}^{2s}}
  \,.
$$

The corresponding [[functional determinant]] is

$$
  \exp(
    E^\prime_{\Delta}(0)
  )
  = (Im \tau)^2 {\vert \eta(\tau)\vert}^4
  \,,
$$

where $\eta$ is the [[Dedekind eta function]].

(recalled e.g. in [Todorov 03, page 3](#Todorov03))

This kind of expression appears as the [[partition function]] of the [[bosonic string]] (e.g. section 6.4.2 in these lectures: [pdf](http://www.damtp.cam.ac.uk/user/tong/string/six.pdf))

## References

* Wikipedia, _[Dedekind eta function](http://en.wikipedia.org/wiki/Dedekind_eta_function)_

* {#Atiyah87} [[Michael Atiyah]], _The logarithm of the Dedekind $\eta$-function_, Math. Ann. 278, 335-380 (1987) ([pdf](http://www.maths.ed.ac.uk/~aar/papers/atiyahlg.pdf))

See also

* {#Todorov03} [[Andrey Todorov]], _The analogue of the Dedekind eta function for CY threefolds_, 2003 [pdf](http://www.ma.huji.ac.il/conf/crelle.pdf)

[[!redirects Dedekind eta functions]]

[[!redirects Dedekind eta-function]]


Dedekind ring

A commutative unital ring $R$ is a __Dedekind ring__ if it is

* Noetherian,
* [[integrally closed ring]],
* and every non-zero [[prime ideal]] in $R$ is maximal.

If $R$ is a Dedekind ring then every ideal can be uniquely factored into prime ideals and the non-zero [[fractional ideal]]s form a group under [[multiplication of ideal]]s.

* Serge Lang, _Algebraic number theory_, GTM 110, Springer 1970, 2000

[[!redirects Dedekind rings]]