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== Group-like structures ==
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Taken from [http://en.wikipedia.org/wiki/Template:Group-like_structures]
 
Taken from [http://en.wikipedia.org/wiki/Template:Group-like_structures]
 
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== Ring-like structures ==
  
 
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|- align="center"
 
|- align="center"
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| style="border-bottom: 2px solid #303060" colspan=11| '''Ring-like structures'''
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| style="border-bottom: 2px solid #303060" colspan=7| '''Ring-like structures'''
 
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|-
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! !! [[Magma]]* !! [[Semigroup]] !! [[Monoid]] !! [[Group]] !! [[Abelian Group]] !! [[Loop]] !! [[Quasigroup]] !! [[Groupoid]] !! [[Cathegory]] !! [[Semicathegory]]
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! !! [[Magma]]* !! [[Semigroup]] !! [[Commutative semigroup]] !! [[Monoid]] !! [[Commutative monoid]] || [[Group]] !! [[Abelian Group]]
 
|-
 
|-
 
! [[Magma (algebra)|Magma]]
 
! [[Magma (algebra)|Magma]]
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| Y || N || N || N || N || N || N || N || N || N
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| [http://www.proofwiki.org/wiki/Definition:Ringoid Ringoid] || ? || ? || ? || ? || ? || ?
 
|-
 
|-
 
! [[Semigroup]]
 
! [[Semigroup]]
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| Y || N || N || N || N || N || N || N || N || N
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| ? || [http://www.proofwiki.org/wiki/Definition:Semiring_%28Abstract_Algebra%29 Semiring] || ? || ? || ? || ? || ?
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|-
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! [[Commutative semigroup]]
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| ? || [http://www.proofwiki.org/wiki/Definition:Additive_Semiring Additive semiring, semiring] || ? || ? || ? || ? || ?
 
|-
 
|-
 
! [[Monoid]]
 
! [[Monoid]]
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| Y || N || N || N || N || N || N || N || N || N
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| ? || [http://www.proofwiki.org/wiki/Definition:Rig Rig, semiring] || ? || ? || ? || ? || ?
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|-
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! [[Monoid|Commutative monoid]]
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| ? || [http://www.proofwiki.org/wiki/Definition:Rig Rig] || ? || ? || ? || ? || ?
 
|-
 
|-
 
! [[Group (mathematics)|Group]]
 
! [[Group (mathematics)|Group]]
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| Y || N || N || N || N || N || N || N || N || N
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| ? || ? || ? || ? || ? || ? || ?
 
|-
 
|-
 
! [[Abelian Group]]
 
! [[Abelian Group]]
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| ? || [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] || ? || ? || ? || ? || ?
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| ? || [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]
−
|-
 
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! [[Loop (algebra)|Loop]]
 
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| Y || N || N || N || N || N || N || N || N || N
 
−
|-
 
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! [[Quasigroup]]
 
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| Y || N || N || N || N || N || N || N || N || N
 
−
|-
 
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! [[Groupoid]]
 
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| Y || N || N || N || N || N || N || N || N || N
 
−
|-
 
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! [[Category (mathematics)|Category]]
 
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| Y || N || N || N || N || N || N || N || N || N
 
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|-
 
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! [[Semicategory]]
 
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| Y || N || N || N || N || N || N || N || N || N
 
 
|}
 
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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
Источник — «http://neerc.ifmo.ru/wiki/index.php?title=Участник:Dgerasimov/Algebra&oldid=33425»