Участник:Dgerasimov/Algebra — различия между версиями
| Строка 14: | Строка 14: | ||
! [[Monoid]] | ! [[Monoid]] | ||
| Y || Y || Y || N || N | | Y || Y || Y || N || N | ||
| + | |- | ||
| + | ! [[Monoid|Commutative monoid]] | ||
| + | | Y || Y || Y || N || Y | ||
|- | |- | ||
! [[Group (mathematics)|Group]] | ! [[Group (mathematics)|Group]] | ||
| Строка 61: | Строка 64: | ||
|- | |- | ||
! [[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] || ? || ? || ? || ? || ? |
|- | |- | ||
! [[Loop (algebra)|Loop]] | ! [[Loop (algebra)|Loop]] | ||
Версия 09:35, 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 | 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 |
| Group | Y | N | N | N | N | N | N | N | N | N |
| Abelian Group | ? | Ring | Unital ring | ? | Commutative ring | ? | ? | ? | ? | ? |
| 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 |
