Group-like algebraic structures

Group-like algebras

                       magma
                   ↙      ↓      ↘
          div           id            ass
         ↙
quasigroup         unital_magma         semigroup
    ↓   +ass ↘   ↙ +inv
    id        
    ↓             ↘
   loop            unverse_semigroup       monoid

          ass           id            inv

                       group

• Magma

• Semigroup

• Monoid

• Group

• Abelian group

• Rack and quandle

• Quasigroup

• loop

• Lie group

• Group theory

Axioms of group-like algebras

• closure, totality

• associativity, associative binary operation

• identity, unique neutral or identity element

• invertability, inverse elements

• divisibility

• commutativity