# nLab invertible magma

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

It is possible to define inverses in a magma without defining an identity element first, yielding a notion of invertible magma

## Definition

A left invertible magma is a magma $(G,(-)\cdot(-):G\times G\to G)$ with a unary operation $(-)^{-1}:G \to G$ called the left inverse or retraction such that

• $a \cdot (b^{-1} \cdot b) = a$
• $(b^{-1} \cdot b) \cdot a = a$

for all $a,b \in G$.

A right invertible magma is a magma $(G,(-)\cdot(-):G\times G\to G)$ with a unary operation $(-)^{-1}:G \to G$ called the right inverse or section such that

• $a \cdot (b \cdot b^{-1}) = a$
• $(b \cdot b^{-1}) \cdot a = a$

for all $a,b \in G$.

An invertible magma is a magma $(G,(-)\cdot(-):G\times G\to G)$ with a unary operation $(-)^{-1}:G \to G$ called the inverse such that

• $a \cdot (b^{-1} \cdot b) = a$
• $(b^{-1} \cdot b) \cdot a = a$
• $a \cdot (b \cdot b^{-1}) = a$
• $(b \cdot b^{-1}) \cdot a = a$

for all $a,b \in G$.

## Properties

Every invertible magma is a cancellative magma?.

The submagma of every power-associative invertible magma $M$ generated by an element $a \in M$ is a cyclic group. This means in particular there is a $\mathbb{Z}$-action on $M$ $(-)^{(-)}:M\times\mathbb{Z}\to M$ called the power.

## Examples

Last revised on June 5, 2021 at 01:51:02. See the history of this page for a list of all contributions to it.