Участник:Dgerasimov/Algebra — различия между версиями

Версия 13:36, 6 ноября 2013

Taken from [1]

	Totality*	Associativity	Identity	Inverses	Commutativity
Group-like structures
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.

	Magma*	Semigroup	Monoid	Commutative monoid	Group	Abelian Group	Loop	Quasigroup	Groupoid	Cathegory	Semicathegory
Ring-like structures
Magma	Y	N	N	?	N	N	N	N	N	N	N
Semigroup	Y	N	N	?	N	N	N	N	N	N	N
Monoid	Y	N	N	?	N	N	N	N	N	N	N
Commutative monoid	?	?	?	?	?	?	?	?	?	?	?
Group	Y	N	N	?	N	N	N	N	N	N	N
Abelian Group	?	Ring	Unital ring	Commutative ring	Division ring (algebra)	Field	?	?	?	?	?
Loop	Y	N	N	?	N	N	N	N	N	N	N
Quasigroup	Y	N	N	?	N	N	N	N	N	N	N
Groupoid	Y	N	N	?	N	N	N	N	N	N	N
Category	Y	N	N	?	N	N	N	N	N	N	N
Semicategory	Y	N	N	?	N	N	N	N	N	N	N

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+== Group-like structures ==
 Taken from [http://en.wikipedia.org/wiki/Template:Group-like_structures]
 {| style="text-align:center"
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 |}
+== Ring-like structures ==
 {| style="text-align:center"