There should be an associative version of a commutative invertible magma. That leads to the concept of a commutative invertible semigroup.
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 $a,b \in G$.
A commutative invertible semigroup is a set $G$ with a binary operation $(-)-(-):G \times G \to G$ called subtraction such that:
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 $a-((b-b)-b)$.
Every abelian group is a commutative invertible semigroup.
The empty magma is a commutative invertible semigroup that is not an abelian group.
invertible semigroup (non-commutative version)
commutative invertible magma (non-associative version)
commutative semigroup? (non-invertible version)
abelian group (unital version)
Last revised on May 25, 2021 at 11:25:08. See the history of this page for a list of all contributions to it.