In classical mathematics, a multiplicatively cancellable rig is a rig $R$ such that each $(r \neq 0) \in R$ is multiplicatively cancellable.

An element $r \in R$ where $R$ is a rig is left multiplicatively cancellable iff for every $s,t \in R$ we have that $(r.s = r.t) \Rightarrow (s = t)$ and it is right multiplicatively cancellable iff for every $s,t \in R$ we have that $(s.r = t.r) \Rightarrow (s = t)$. An element $r \in R$ is multiplicatively cancellable iff it is left multiplicatively cancellable and right multiplicatively cancellable.

If $R$ is a commutative rig, an element is left multiplicatively cancellable iff it is right multiplicatively cancellable.

A residue multiplicatively cancellable rig is a rig $R$ such that for all $r \in R$, if $r$ is not multiplicatively cancellable, then $r = 0$.

Definition

A Heyting multiplicatively cancellable rig is a rig $R$ with a tight apartness relation$\#$ such that each $(r \# 0) \in R$ is multiplicatively cancellable.

Definition

A discrete multiplicatively cancellable rig is a rig $R$ such that each $r \in R$ is zero xor multiplicatively cancellable.

Last revised on August 3, 2022 at 20:28:29.
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