nLab invertible magma

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

• Every group is an invertible magma.

• Every invertible semigroup and nonassociative group is an invertible magma.

Related concepts

• magma (noninvertible version)

• invertible unital magma (unital version)

• commutative invertible magma (commutative version)

• invertible semigroup (associative version)

• invertible quasigroup (divisible version)

• reciprocal algebra

• centipede mathematics