commutative invertible semigroup




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×GG)(G,(-)+(-):G\times G\to G) with a unary operation :GG-:G \to G called the inverse such that a+b+(b)=aa + b + (-b) = a for all a,bGa,b \in G.

With subtraction only

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

  • For all aa and bb in GG, aa=bba-a=b-b
  • For all aa in GG, (aa)((aa)a)=a(a-a)-((a-a)-a)=a
  • For all aa and bb in GG, a((bb)b)=b((aa)a)a-((b-b)-b) = b-((a-a)-a)
  • For all aa, bb, and cc in GG, a(bc)=(a((cc)c)ba-(b-c)=(a-((c-c)-c)-b

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

Last revised on May 25, 2021 at 11:25:08. See the history of this page for a list of all contributions to it.