nLab commutative invertible semigroup

Idea

There should be an associative version of a commutative invertible magma. That leads to the concept of a commutative invertible semigroup.

Definition

A commutative invertible semigroup is a commutative semigroup $(G,(-)+(-):G\times G\to G)$ with a unary operation $-:G \to G$ called the inverse such that $a + b + (-b) = a$ for all $a,b \in G$.

With subtraction only

A commutative invertible semigroup is a set $G$ with a binary operation $(-)-(-):G \times G \to G$ called subtraction such that:

• For all $a$ and $b$ in $G$, $a-a=b-b$
• For all $a$ in $G$, $(a-a)-((a-a)-a)=a$
• For all $a$ and $b$ in $G$, $a-((b-b)-b) = b-((a-a)-a)$
• For all $a$, $b$, and $c$ in $G$, $a-(b-c)=(a-((c-c)-c)-b$

For any element $a$ in a commutative invertible semigroup $G$, the element $a-a$ is called an identity element, and the element $(a-a)-a$ is called the inverse element of $a$. For all elements $a$ and $b$, addition of $a$ and $b$ is defined as $a-((b-b)-b)$.

Properties

• Every commutative invertible semigroup is a commutative quasigroup.

Examples

• Every abelian group is a commutative invertible semigroup.

• The empty magma is a commutative invertible semigroup that is not an abelian group.

• Every ring is a monoid object in commutative invertible semigroups.

• Similarly, every (non-associative) unital $\mathbb{Z}$-algebra is an H-space object in commutative invertible semigroups.

Related concepts

• invertible semigroup (non-commutative version)

• commutative invertible magma (non-associative version)

• commutative semigroup (non-invertible version)

• abelian group (unital version)

• pointed commutative invertible semigroup

• Tarski group