# nLab absorption monoid

Contents

### Context

#### Algebra

higher algebra

universal algebra

## Theorems

#### Monoid theory

monoid theory in algebra:

# Contents

## Definition

An absorption monoid or annihilation monoid is a monoid $(M,1,\cdot)$ that is also an absorption magma $(M,0)$.

## Properties

### Zero divisors

A non-zero element $a \in M$ is a zero divisor if there exists a non-zero element $b \in M$ such that $a \cdot b = 0$.

### Localization and group completion

The localization of an absorption monoid at $0$ is the trivial group. Because of this, the group completion of any absorption monoid is the trivial group. This is why one speaks of division monoids instead of groups in the context of absorption monoids, and in particular, why the additive identity element in any nontrivial field has no multiplicative inverse.

## Examples

Last revised on May 20, 2022 at 21:17:38. See the history of this page for a list of all contributions to it.