Comment by fyp

Most algebraic structures are best understood by which axioms it satisfies. For example basically every subset of axioms of an abelian group is useful enough to have a name. Wiki has a really nice table:

Semigroupoid

Small Category

Groupoid

Magma

Quasigroup

Unital Magma

Loop

Semigroup

Inverse Semigroup

Monoid

Commutative monoid

Group

Abelian group

https://en.wikipedia.org/wiki/Abelian_group