Участник:Dgerasimov/Algebra — различия между версиями

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(Ring-like structures)
(Ring-like structures)
 
Строка 55: Строка 55:
 
|-
 
|-
 
! [[Magma (algebra)|Magma]]
 
! [[Magma (algebra)|Magma]]
| [http://www.proofwiki.org/wiki/Definition:Ringoid Ringoid] || ? || N || N || ? || N || N
+
| [http://www.proofwiki.org/wiki/Definition:Ringoid Ringoid] || ? || ? || ? || ? || ? || ?
 
|-
 
|-
 
! [[Semigroup]]
 
! [[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]]
 
! [[Commutative semigroup]]
Строка 64: Строка 64:
 
|-
 
|-
 
! [[Monoid]]
 
! [[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]]
 
! [[Monoid|Commutative monoid]]
Строка 70: Строка 70:
 
|-
 
|-
 
! [[Group (mathematics)|Group]]
 
! [[Group (mathematics)|Group]]
| Y || ? || N || N || ? || N || N
+
| ? || ? || ? || ? || ? || ? || ?
 
|-
 
|-
 
! [[Abelian Group]]
 
! [[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]
 
| ? || [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]
 
|}
 
|}

Текущая версия на 14:00, 6 ноября 2013

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