Участник:Dgerasimov/Algebra — различия между версиями

Текущая версия на 14:00, 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	Commutative semigroup	Monoid	Commutative monoid	Group	Abelian Group
Ring-like structures
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

@@ Строка 55: / Строка 55: @@
 |-
 ! [[Magma (algebra)|Magma]]
-| [http://www.proofwiki.org/wiki/Definition:Ringoid Ringoid] || ? || N || N || ? || N || N
+| [http://www.proofwiki.org/wiki/Definition:Ringoid Ringoid] || ? || ? || ? || ? || ? || ?
 |-
 ! [[Semigroup]]
-| Y || [http://www.proofwiki.org/wiki/Definition:Semiring_%28Abstract_Algebra%29 Semiring] || ? || N || ? || N || N
+| ? || [http://www.proofwiki.org/wiki/Definition:Semiring_%28Abstract_Algebra%29 Semiring] || ? || ? || ? || ? || ?
 |-
 ! [[Commutative semigroup]]
@@ Строка 64: / Строка 64: @@
 |-
 ! [[Monoid]]
-| Y || [http://www.proofwiki.org/wiki/Definition:Rig Rig, semiring] || N || N || ? || N || N
+| ? || [http://www.proofwiki.org/wiki/Definition:Rig Rig, semiring] || ? || ? || ? || ? || ?
 |-
 ! [[Monoid|Commutative monoid]]
@@ Строка 70: / Строка 70: @@
 |-
 ! [[Group (mathematics)|Group]]
-| Y || ? || N || N || ? || N || N
+| ? || ? || ? || ? || ? || ? || ?
 |-
 ! [[Abelian Group]]
 | ? || [http://en.wikipedia.org/wiki/Ring_%28mathematics%29 Ring] || ? || [http://en.wikipedia.org/wiki/Ring_%28mathematics%29 Unital ring] || [http://en.wikipedia.org/wiki/Commutative_ring Commutative ring] || [http://mathworld.wolfram.com/DivisionAlgebra.html Division ring (algebra)] || [http://en.wikipedia.org/wiki/Field_%28mathematics%29 Field]
 |}