nLab cancellative binary function

Idea

A version of cancellative magmas where the binary function’s codomain need not coincide with its domain set. This allows the concept to be generalised from magmas and monoids to other binary functions such as actions and modules.

Definition

Given sets $A$, $B$, and $C$, a binary function $f:A \times B \to C$ is left cancellative if for all $c \in B$ and $d \in B$, if $a \cdot c = a \cdot d$ for all $a \in A$, then $c = d$, and it is right cancellative if for all $a \in A$ and $b \in A$, if $a \cdot c = b \cdot c$ for all $c \in A$, then $a = b$. It is cancellative if it is both left cancellative and right cancellative.

See also

• cancellative monoid
• torsion-free module
• integral domain