Участник:Dgerasimov/Algebra

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