Участник:Dgerasimov/Algebra — различия между версиями
(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"...») |
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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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| + | {| style="text-align:center" | ||
| + | |- align="center" | ||
| + | | style="border-bottom: 2px solid #303060" colspan=11| '''Ring-like structures''' | ||
| + | |- | ||
| + | ! !! [[Magma]]* !! [[Semigroup]] !! [[Monoid]] !! [[Group]] !! [[Abelian Group]] !! [[Loop]] !! [[Quasigroup]] !! [[Groupoid]] !! [[Cathegory]] !! [[Semicathegory]] | ||
| + | |- | ||
| + | ! [[Magma (algebra)|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 (mathematics)|Group]] | ||
| + | | Y || N || N || N || N || N || N || N || N || N | ||
| + | |- | ||
| + | ! [[Abelian Group]] | ||
| + | | Y || N || N || N || N || N || N || N || N || N | ||
| + | |- | ||
| + | ! [[Loop (algebra)|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 (mathematics)|Category]] | ||
| + | | Y || N || N || N || N || N || N || N || N || N | ||
| + | |- | ||
| + | ! [[Semicategory]] | ||
| + | | Y || N || N || N || N || N || N || N || N || N | ||
|} | |} | ||
Версия 09:22, 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 |
| 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 | Y | N | N | N | N | N | N | N | N | N |
| 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 |
