Участник:Dgerasimov/Algebra — различия между версиями

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(Новая страница: «Taken from [http://en.wikipedia.org/wiki/Template:Group-like_structures] {| style="text-align:center" |- align="center" | style="border-bottom: 2px solid #303060"...»)
 
(→‎Ring-like structures)
 
(не показано 9 промежуточных версий этого же участника)
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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]
 
{| style="text-align:center"
 
{| style="text-align:center"
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! [[Monoid]]
 
! [[Monoid]]
 
| Y || Y || Y || N || N
 
| Y || Y || Y || N || N
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|-
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! [[Monoid|Commutative monoid]]
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| Y || Y || Y || N || Y
 
|-
 
|-
 
! [[Group (mathematics)|Group]]
 
! [[Group (mathematics)|Group]]
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|-
 
|-
 
| || colspan="5" | <small>**Each element of a [[Loop (algebra)|loop]] has a left and right inverse, but these need not coincide.</small>
 
| || colspan="5" | <small>**Each element of a [[Loop (algebra)|loop]] has a left and right inverse, but these need not coincide.</small>
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|}
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== Ring-like structures ==
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{| style="text-align:center"
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|- align="center"
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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]] !! [[Commutative semigroup]] !! [[Monoid]] !! [[Commutative monoid]] || [[Group]] !! [[Abelian Group]]
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|-
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! [[Magma (algebra)|Magma]]
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| [http://www.proofwiki.org/wiki/Definition:Ringoid Ringoid] || ? || ? || ? || ? || ? || ?
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|-
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! [[Semigroup]]
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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] || ? || ? || ? || ? || ?
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|-
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! [[Monoid]]
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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] || ? || ? || ? || ? || ?
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|-
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! [[Group (mathematics)|Group]]
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| ? || ? || ? || ? || ? || ? || ?
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|-
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! [[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] || [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
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