# nLab cancellative monoid

Contents

### Context

#### Algebra

higher algebra

universal algebra

## Theorems

#### Monoid theory

monoid theory in algebra:

# Contents

## Definition

### In set theory

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.