Ladder of Algebraic Structures
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:SemigroupoidSmall CategoryGroupoidMagmaQuasigroupUnital MagmaLoopSemigroupInverse SemigroupMonoidCommutative monoidGroupAbelian grouphttps://en.wikipedia.org/wiki/Abelian_group