Участник:Dgerasimov/Algebra — различия между версиями
| Строка 49: | Строка 49: | ||
| style="border-bottom: 2px solid #303060" colspan=11| '''Ring-like structures''' | | style="border-bottom: 2px solid #303060" colspan=11| '''Ring-like structures''' | ||
|- | |- | ||
| − | ! !! [[Magma]]* !! [[Semigroup]] !! [[Monoid]] !! [[Group]] !! [[Abelian Group]] !! [[Loop]] !! [[Quasigroup]] !! [[Groupoid]] !! [[Cathegory]] !! [[Semicathegory]] | + | ! !! [[Magma]]* !! [[Semigroup]] !! [[Monoid]] !! [[Commutative monoid]] || [[Group]] !! [[Abelian Group]] !! [[Loop]] !! [[Quasigroup]] !! [[Groupoid]] !! [[Cathegory]] !! [[Semicathegory]] |
|- | |- | ||
! [[Magma (algebra)|Magma]] | ! [[Magma (algebra)|Magma]] | ||
| − | | Y || N || N || N || N || N || N || N || N || N | + | | Y || N || N || ? || N || N || N || N || N || N || N |
|- | |- | ||
! [[Semigroup]] | ! [[Semigroup]] | ||
| − | | Y || N || N || N || N || N || N || N || N || N | + | | Y || N || N || ? || N || N || N || N || N || N || N |
|- | |- | ||
! [[Monoid]] | ! [[Monoid]] | ||
| − | | Y || N || N || N || N || N || N || N || N || N | + | | Y || N || N || ? || N || N || N || N || N || N || N |
| + | |- | ||
| + | ! [[Monoid|Commutative monoid]] | ||
| + | | ? || ? || ? || ? || ? || ? || ? || ? || ? || ? || ? | ||
|- | |- | ||
! [[Group (mathematics)|Group]] | ! [[Group (mathematics)|Group]] | ||
| − | | Y || N || N || N || N || N || N || N || N || N | + | | Y || N || N || ? || N || N || N || 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/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] || ? || ? || ? || ? || ? |
|- | |- | ||
! [[Loop (algebra)|Loop]] | ! [[Loop (algebra)|Loop]] | ||
| − | | Y || N || N || N || N || N || N || N || N || N | + | | Y || N || N || ? || N || N || N || N || N || N || N |
|- | |- | ||
! [[Quasigroup]] | ! [[Quasigroup]] | ||
| − | | Y || N || N || N || N || N || N || N || N || N | + | | Y || N || N || ? || N || N || N || N || N || N || N |
|- | |- | ||
! [[Groupoid]] | ! [[Groupoid]] | ||
| − | | Y || N || N || N || N || N || N || N || N || N | + | | Y || N || N || ? || N || N || N || N || N || N || N |
|- | |- | ||
! [[Category (mathematics)|Category]] | ! [[Category (mathematics)|Category]] | ||
| − | | Y || N || N || N || N || N || N || N || N || N | + | | Y || N || N || ? || N || N || N || N || N || N || N |
|- | |- | ||
! [[Semicategory]] | ! [[Semicategory]] | ||
| − | | Y || N || N || N || N || N || N || N || N || N | + | | Y || N || N || ? || N || N || N || N || N || N || N |
|} | |} | ||
Версия 12:20, 6 ноября 2013
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 | |||||||||||
| Magma* | Semigroup | Monoid | Commutative monoid | Group | Abelian Group | Loop | Quasigroup | Groupoid | Cathegory | Semicathegory | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Magma | Y | N | N | ? | N | N | N | N | N | N | N |
| Semigroup | Y | N | N | ? | N | N | N | N | N | N | N |
| Monoid | Y | N | N | ? | N | N | N | N | N | N | N |
| Commutative monoid | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? |
| Group | Y | N | N | ? | N | N | N | N | N | N | N |
| Abelian Group | ? | Ring | Unital ring | Commutative ring | Division ring (algebra) | Field | ? | ? | ? | ? | ? |
| Loop | Y | N | N | ? | N | N | N | N | N | N | N |
| Quasigroup | Y | N | N | ? | N | N | N | N | N | N | N |
| Groupoid | Y | N | N | ? | N | N | N | N | N | N | N |
| Category | Y | N | N | ? | N | N | N | N | N | N | N |
| Semicategory | Y | N | N | ? | N | N | N | N | N | N | N |
