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Group-like structures

Taken from [1]

Group-like structures
Totality* Associativity Identity Inverses Commutativity
Magma Y N N N N
Semigroup Y Y N N N
Monoid Y Y Y N N
Commutative monoid Y Y Y N Y
Group Y Y Y Y N
Abelian Group Y Y Y Y Y
Loop Y N Y Y** N
Quasigroup Y N N N N
Groupoid N Y Y Y N
Category N Y Y N N
Semicategory N Y N N N
*Closure, which is used in many sources to define group-like structures, is an equivalent axiom to totality, though defined differently.
**Each element of a loop has a left and right inverse, but these need not coincide.

Ring-like structures

Ring-like structures
Magma* Semigroup Commutative semigroup Monoid Commutative monoid Group Abelian Group
Magma Ringoid  ?  ?  ?  ?  ?  ?
Semigroup  ? Semiring  ?  ?  ?  ?  ?
Commutative semigroup  ? Additive semiring, semiring  ?  ?  ?  ?  ?
Monoid  ? Rig, semiring  ?  ?  ?  ?  ?
Commutative monoid  ? Rig  ?  ?  ?  ?  ?
Group  ?  ?  ?  ?  ?  ?  ?
Abelian Group  ? Ring  ? Unital ring Commutative ring Division ring (algebra) Field