Editing Magma (algebra) (section)

== Types of magma ==
[[Image:Magma to group4.svg|thumb|right|300px|Algebraic structures from magmas to groups]]
Magmas are not often studied as such; instead there are several different kinds of magma, depending on what axioms the operation is required to satisfy. Commonly studied types of magma include:
* '''[[Quasigroup]]''': A magma where [[division (mathematics)|division]] is always possible.
** [[Loop (algebra)|'''Loop''']]: A quasigroup with an [[identity element]].
* '''[[Semigroup]]''': A magma where the operation is [[associative]].
** '''[[Monoid]]''': A semigroup with an identity element.
* [[Group (mathematics)|'''Group''']]: A magma with inverse, associativity, and an identity element.

Note that each of divisibility and invertibility imply the [[cancellation property]].

; Magmas with [[commutative|commutativity]]:
* '''[[Commutative magma]]''': A magma with commutativity.
* '''[[Commutative monoid]]''': A monoid with commutativity.
* '''[[Abelian group]]''': A group with commutativity.