[[!redirects Sandbox > history]] [[!redirects Sandbox]] < [[nlab:Sandbox]] ## Number line rigs We want properties which hold for the natural numbers and the Dedekind real numbers. A rig is a set $S$ with elements $0:S$ and $1:S$ and functions $(-)+(-):S \times S \to S$ and $(-)\cdot(-):S \times S \to S$ such that $(S, 0, +)$ is a commutative monoid, $(S, 1, \cdot)$ is a commutative monoid, and $(-)\cdot(-)$ left and right distributes over $(-)+(-)$. A rig is ordered if it comes with a linear order $\lt$ and additionally $0 \lt 1$, and if $0 \lt a$ and $0 \lt b$, then $0 \lt a + b$ and $0 \lt a \cdot b$. A rig is an ordered l-rig if additionally, the negation of the linear order $a \leq b \coloneqq \neg (b \lt a)$ is a pseudolattice with functions $\min:S \times S \to S$ and $\max:S \times S \to S$. An ordered l-rig is an ordered integral l-rig if addition is cancellative, and the multiplicatively regular elements in $S$ are either greater than zero or less than zero. A metric is a function $\rho:S \times S \to S_{\geq 0}$ to the non-negative values in $S$ such that for all elements $a:S$, $b:S$, and $c:S$, $\rho(a, b) + \min(a, b) = \max(a, b)$. This can be shown to be a metric: * $a = b \iff \rho(a, b) = 0$ * $\rho(a, b) = \rho(b, a)$ * $\rho(a, b) \leq \rho(a, c) + \rho(c, b)$ Thus, it is an ordered metric integral l-rig. An ordered metric integral l-rig is Archimedean if it satisfies the Archimedean property: for every element $0 \lt a$, there is a positive natural number $n$ such that $a \lt n$, and there is a positive natural number $m$ such that $1 \lt m \cdot a$. A number line rig is an Archimedean ordered metric integral l-rig. The natural numbers are the initial Anumber line rig, and the Dedekind real numbers are the terminal number line rig. One could construct the Grothendieck group to get an number line ring, and construct the ring of fractions to get an number line field. category: redirected to nlab