# Homotopy Type Theory Sandbox > history (Rev #73, changes)

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# Contents

## Commutative algebra

### Trivial ring

A commutative ring $R$ is trivial if $0 = 1$.

### Discrete rings

A commutative ring $R$ is discrete if for all elements $x:R$ and $y:R$, either $x = y$, or $x = y$ implies $0 = 1$.

$\prod_{x:R} \prod_{y:R} (x = y) + (x = y) \to (0 = 1)$

### Regular elements

Given a commutative ring $R$, a term $e:R$ is left regular if for all $a:R$ and $b:R$, $e \cdot a = e \cdot b$ implies $a = b$.

$\mathrm{isLeftRegular}(e) \coloneqq \prod_{a:R} \prod_{b:R} (e \cdot a = e \cdot b) \to (a = b)$

A term $e:R$ is right regular if for all $a:R$ and $b:R$, $a \cdot e = b \cdot e$ implies $a = b$.

$\mathrm{isRightRegular}(e) \coloneqq \prod_{a:R} \prod_{b:R} (a \cdot e = b \cdot e) \to (a = b)$

An term $e:R$ is regular if it is both left regular and right regular.

$\mathrm{isRegular}(e) \coloneqq \mathrm{isLeftRegular}(e) \times \mathrm{isRightRegular}(e)$

The multiplicative monoid of regular elements in $R$ is the submonoid of all regular elements in $R$

$\mathrm{Reg}(R) \coloneqq \sum_{e:R} \mathrm{isRegular}(e)$

### Non-regular elements and integral rings

An element $x:R$ is non-regular if $x$ being regular implies that $0 = 1$

$\mathrm{isNonRegular}(x) \coloneqq \mathrm{isRegular}(x) \to (0 = 1)$

Zero is always a non-regular element of $R$. By definition of non-regular, if $0$ is regular, then the ring $R$ is trivial.

A commutative ring is integral if the type of all non-regular elements in $R$ is contractible:

$\mathrm{isContr}\left(\sum_{x:R} \mathrm{isNonRegular}(x)\right)$

### Invertible elements

Given a commutative ring $R$, a term $e:R$ is left invertible if the fiber of right multiplication by $e$ at $1$ is inhabited.

$\mathrm{isLeftInvertible}(e) \coloneqq \left[\sum_{a:R} a \cdot e = 1\right]$

A term $e:R$ is right invertible if the fiber of left multiplication by $e$ at $1$ is inhabited.

$\mathrm{isRightInvertible}(e) \coloneqq \left[\sum_{a:R} e \cdot a = 1\right]$

An term $e:R$ is invertible if it is both left invertible and right invertible.

$\mathrm{isInvertible}(e) \coloneqq \mathrm{isLeftInvertible}(e) \times \mathrm{isRightInvertible}(e)$

The multiplicative group of invertible elements in $R$ is the subgroup of all invertible elements in $R$

$R^\times \coloneqq \sum_{e:R} \mathrm{isInvertible}(e)$

### Non-regular Non-invertible and non-invertible elements and fields

An element $x:R$ is non-regular non-invertible if$x$ being regular invertible implies that$0 = 1$

 \mathrm{isNonRegular}(x) \mathrm{isNonInvertible}(x) \coloneqq \mathrm{isRegular}(x) \mathrm{isInvertible}(x) \to (0 = 1)

An Zero is always a non-invertible element of x:R R . is By non-invertible definition of non-invertible, if x 0 being is invertible invertible, implies then that the ring 0 R = 1 is trivial.

$\mathrm{isNonInvertible}(x) \coloneqq \mathrm{isInvertible}(x) \to (0 = 1)$

A commutative ring is a field if the type of all non-invertible elements in $R$ is contractible:

Zero is always a non-regular and non-invertible element of $R$. By definition of non-regular and non-invertible, if $0$ is regular or invertible, then the ring $R$ is trivial.

$\mathrm{isContr}\left(\sum_{x:R} \mathrm{isNonInvertible}(x)\right)$

A commutative ring is integral if every non-regular element is equal to zero. A commutative ring is a field if every non-invertible element is equal to zero.

### References

• Frank Quinn, Proof Projects for Teachers (pdf)

• Henri Lombardi, Claude Quitté, Commutative algebra: Constructive methods (Finite projective modules) (arXiv:1605.04832)

## Real numbers

### Pointed abelian groups

A pointed abelian group is a set $A$ with an element $1:A$ called one and a binary operation $(-)-(-):A \times A \to A$ called subtraction such that

• for all elements $a:A$, $a - a = 1 - 1$

• for all elements $a:A$, $(1 - 1) - ((1 - 1) - a) = a$

• for all elements $a:A$ and $b:A$, $a - ((1 - 1) - b) = b - ((1 - 1) - a)$

• for all elements $a:A$, $b:A$, and $c:A$, $a - (b - c) = (a - ((1 - 1) - c)) - b$

### Pointed halving groups

A pointed halving group is a pointed abelian group G with a function $(-)/2:G \to G$ called halving or dividing by two such that for all elements $g:G$, $g/2 = g - g/2$.

### Totally ordered pointed halving groups

A pointed halving group $R$ is a totally ordered pointed halving group if it comes with a function $\max:R \times R \to R$ such that

• for all elements $a:R$, $\max(a, a) = a$

• for all elements $a:R$ and $b:R$, $\max(a, b) = \max(b, a)$

• for all elements $a:R$, $b:R$, and $c:R$, $\max(a, \max(b, c)) = \max(\max(a, b), c)$

• for all elements $a:R$ and $b:R$, $\max(a, b) = a$ or $\max(a, b) = b$

• for all elements $a:R$ and $b:R$, $\max(a, b) = b$ implies that for all elements $c:R$, $\max(a - c, b - c) = b - c$

### Strictly ordered pointed halving groups

A totally ordered pointed halving group $R$ is a strictly ordered pointed abelian group if it comes with a type family $\lt$ such that

• for all elements $a:R$ and $b:R$, $a \lt b$ is a proposition
• for all elements $a:R$, $a \lt a$ is false
• for all elements $a:R$, $b:R$, and $c:R$, if $a \lt c$, then $a \lt b$ or $b \lt c$
• for all elements $a:R$ and $b:R$, if $a \lt b$ is false and $b \lt a$ is false, then $a = b$
• for all elements $a:R$ and $b:R$, if $a \lt b$, then $b \lt a$ is false.
• $1 - 1 \lt 1$
• for all elements $a:R$ and $b:R$, if $1 - 1 \lt a$ and $1 - 1 \lt b$, then $(1 - 1) \lt a - ((1 - 1) - b)$

The homotopy initial strictly ordered pointed halving group is the dyadic rational numbers $\mathbb{D}$.

### Archimedean ordered pointed halving groups

A strictly ordered pointed halving group $A$ is an Archimedean ordered pointed halving group if for all elements $a:A$ and $b:A$, if $a \lt b$, then there merely exists a dyadic rational number $d:\mathbb{D}$ such that $a \lt h(d)$ and $h(d) \lt b$.

### Sequentially Cauchy complete Archimedean ordered pointed halving groups

Let $A$ be an Archimedean ordered pointed halving group and let

$A_{+} \coloneqq \sum_{a:A} 1 - 1 \lt a$

be the positive elements in $A$.

A sequence in $A$ is a function $x:\mathbb{N} \to A$.

A sequence $x:\mathbb{N} \to A$ is a Cauchy sequence if for all positive elements $\epsilon:A_{+}$, there merely exists a natural number $N:\mathbb{N}$ such that for all natural numbers $i:\mathbb{N}$ and $j:\mathbb{N}$ such that $N \leq i$ and $N \leq j$, $\max(x_i - x_j, x_j - x_i) \lt \epsilon$.

An element $l:A$ is said to be a limit of the sequence $x:\mathbb{N} \to A$ if for all positive elements $\epsilon:A_{+}$, there merely exists a natural number $N:\mathbb{N}$ such that for all natural numbers $i:\mathbb{N}$ such that $N \leq i$, $\max(x_i - l, l - x_i) \lt \epsilon$

$A$ is sequentially Cauchy complete if every Cauchy sequence in $A$ merely has a limit.

Revision on June 19, 2022 at 22:30:53 by Anonymous?. See the history of this page for a list of all contributions to it.