# nLab invertible quasigroup

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

A quasigroup with a two-sided inverse

## Definition

An invertible quasigroup is a quasigroup $(G,\cdot,\backslash,/)$ 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 $a,b \in G$.

### Without division

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

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

and

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

for all $a,b \in G$.

## Examples

Last revised on June 14, 2021 at 10:58:17. See the history of this page for a list of all contributions to it.