A monoid$(A, \cdot, 1)$ is called left cancellative if

$\underset{a,b,z \in A}{\forall}
\left(
\left(
z \cdot a = z \cdot b
\right)
\Rightarrow
\left(
a = b
\right)
\right)$

and called right cancellative if

$\underset{a,b,z \in A}{\forall}
\left(
\left(
a \cdot z = b \cdot z
\right)
\Rightarrow
\left(
a = b
\right)
\right)$

It is called cancellative if it is both left cancellative and right cancellative.

In infinity-groupoid theory

A monoid ($0$-truncated$A_3$-space) $(A, \cdot, 1)$ is called left cancellative if for all objects $a \in A$ and $b \in A$ the homotopy fiber of the functor $L:A \to A$, defined as $L_a(z) \coloneqq a \cdot z$, at $b$ is $(-1)$-truncated, and is called right cancellative if for all elements $a \in A$ and $b \in B$ the homotopy fiber of the functor $R:A \to A$, defined as $R_a(z) \coloneqq z \cdot a$ at $b$ is $(-1)$-truncated. It is called cancellative if it is both left cancellative and right cancellative.