nLab invertible quasigroup

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

• Every group is an invertible quasigroup.

• Every associative quasigroup and nonassociative group is an invertible quasigroup.

Related concepts

• quasigroup (noninvertible version)

• nonassociative group (unital version)

• commutative invertible quasigroup (commutative version)

• associative quasigroup (associative version)

• invertible magma (nondivisible version)

• Malcev variety

• centipede mathematics