Homotopy Type Theory Sandbox (Rev #68, changes)

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Euclidean semirings

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

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

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

The multiplicative submonoid of cancellative elements in $R$ is the subset of all cancellative elements in $R$

A Euclidean semiring is a additively cancellative commutative semiring $R$ for which there exists a function $d \colon \mathrm{Can}(R) \to \mathbb{N}$ from the multiplicative submonoid of cancellative elements in $R$ to the natural numbers, often called a degree function, a function $(-)\div(-):R \times \mathrm{Can}(R) \to R$ called the division function, and a function $(-)\;\%\;(-):R \times \mathrm{Can}(R) \to R$ called the remainder function, such that for all $a \in R$ and $b \in \mathrm{Can}(R)$, $a = (a \div b) \cdot b + (a\;\%\; b)$ and either $a\;\%\; b = 0$ or $d(a\;\%\; b) \lt d(g)$.

Non-cancellative and non-invertible elements

Given a ring $R$, an element $x \in R$ is non-cancellative if: if there is an element $y \in \mathrm{Can}(R)$ with injection $i:\mathrm{Can}(R) \to R$ such that $i(y) = x$, then $0 = 1$. An element $x \in R$ is non-invertible if: if there is an element $y \in R^\times$ with injection $j:R^\times \to R$ such that $j(y) = x$, then $0 = 1$.

$A_n$ abelian groups

$A_1$-abelian groups are pointed objects in abelian groups

$A_2$-abelian groups are unital magma objects in abelian groups

$A_3$-abelian groups are monoid objects in abelian groups