Участник:Dgerasimov/Algebra — различия между версиями
(→Ring-like structures) |
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| − | | style="border-bottom: 2px solid #303060" colspan= | + | | style="border-bottom: 2px solid #303060" colspan=7| '''Ring-like structures''' |
|- | |- | ||
| − | ! !! [[Magma]]* !! [[Semigroup]] !! [[Monoid]] !! [[Commutative monoid]] || [[Group]] !! [[Abelian Group | + | ! !! [[Magma]]* !! [[Semigroup]] !! [[Commutative semigroup]] !! [[Monoid]] !! [[Commutative monoid]] || [[Group]] !! [[Abelian Group]] |
|- | |- | ||
! [[Magma (algebra)|Magma]] | ! [[Magma (algebra)|Magma]] | ||
| − | | | + | | [http://www.proofwiki.org/wiki/Definition:Ringoid Ringoid] || ? || N || N || ? || N || N |
|- | |- | ||
! [[Semigroup]] | ! [[Semigroup]] | ||
| − | | Y || | + | | Y || [http://www.proofwiki.org/wiki/Definition:Semiring_%28Abstract_Algebra%29 Semiring] || ? || N || ? || N || N |
| + | |- | ||
| + | ! [[Commutative semigroup]] | ||
| + | | ? || ? || ? || ? || ? || ? || ? | ||
|- | |- | ||
! [[Monoid]] | ! [[Monoid]] | ||
| − | | Y | + | | Y || ? || N || N || ? || N || N |
|- | |- | ||
! [[Monoid|Commutative monoid]] | ! [[Monoid|Commutative monoid]] | ||
| − | + | | ? || ? || ? || ? || ? || ? || ? | |
|- | |- | ||
! [[Group (mathematics)|Group]] | ! [[Group (mathematics)|Group]] | ||
| − | | Y | + | | 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] |
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Версия 13:45, 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 | ? | N | N | ? | N | N |
| Semigroup | Y | Semiring | ? | N | ? | N | N |
| Commutative semigroup | ? | ? | ? | ? | ? | ? | ? |
| Monoid | Y | ? | N | N | ? | N | N |
| Commutative monoid | ? | ? | ? | ? | ? | ? | ? |
| Group | Y | ? | N | N | ? | N | N |
| Abelian Group | ? | Ring | ? | Unital ring | Commutative ring | Division ring (algebra) | Field |
