Difference between revisions of "Template:Group-like structures"
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| − | | style="border-bottom: 2px solid #303060" colspan=6| '''Group-like structures. The entries say whether the property is ''required''.''' | + | | style="border-bottom: 2px solid #303060; text-align:center;" colspan=6 | '''Group-like structures. The entries say whether the property is ''required''.''' |
|- | |- | ||
! !! [[Total Function|Totality]]* !! [[Associativity]] !! [[Identity element|Identity]] !! [[Inverse element|Divisibility]] !! [[Commutativity]] | ! !! [[Total Function|Totality]]* !! [[Associativity]] !! [[Identity element|Identity]] !! [[Inverse element|Divisibility]] !! [[Commutativity]] | ||
|- | |- | ||
! [[Semicategory]] | ! [[Semicategory]] | ||
| − | | {{no}} || {{yes}} || {{no}} || {{no}} || {{no}} | + | | {{no|Unneeded}} || {{yes|Required}} || {{no|Unneeded}} || {{no|Unneeded}} || {{no|Unneeded}} |
|- | |- | ||
! [[Category (mathematics)|Category]] | ! [[Category (mathematics)|Category]] | ||
| − | | {{no}} || {{yes}} || {{yes}} || {{no}} || {{no}} | + | | {{no|Unneeded}} || {{yes|Required}} || {{yes|Required}} || {{no|Unneeded}} || {{no|Unneeded}} |
|- | |- | ||
! [[Groupoid]] | ! [[Groupoid]] | ||
| − | | {{no}} || {{yes}} || {{yes}} || {{yes}} || {{no}} | + | | {{no|Unneeded}} || {{yes|Required}} || {{yes|Required}} || {{yes|Required}} || {{no|Unneeded}} |
|- | |- | ||
! [[Magma (algebra)|Magma]] | ! [[Magma (algebra)|Magma]] | ||
| − | | {{yes}} || {{no}} || {{no}} || {{no}} || {{no}} | + | | {{yes|Required}} || {{no|Unneeded}} || {{no|Unneeded}} || {{no|Unneeded}} || {{no|Unneeded}} |
|- | |- | ||
! [[Quasigroup]] | ! [[Quasigroup]] | ||
| − | | {{yes}} || {{no}} || {{no}} || {{yes}} || {{no}} | + | | {{yes|Required}} || {{no|Unneeded}} || {{no|Unneeded}} || {{yes|Required}} || {{no|Unneeded}} |
|- | |- | ||
! [[Loop (algebra)|Loop]] | ! [[Loop (algebra)|Loop]] | ||
| − | | {{yes}} || {{no}} || {{yes}} || {{yes}} || {{no}} | + | | {{yes|Required}} || {{no|Unneeded}} || {{yes|Required}} || {{yes|Required}} || {{no|Unneeded}} |
|- | |- | ||
! [[Semigroup]] | ! [[Semigroup]] | ||
| − | | {{yes}} || {{yes}} || {{no}} || {{no}} || {{no}} | + | | {{yes|Required}} || {{yes|Required}} || {{no|Unneeded}} || {{no|Unneeded}} || {{no|Unneeded}} |
|- | |- | ||
! [[Monoid]] | ! [[Monoid]] | ||
| − | | {{yes}} || {{yes}} || {{yes}} || {{no}} || {{no}} | + | | {{yes|Required}} || {{yes|Required}} || {{yes|Required}} || {{no|Unneeded}} || {{no|Unneeded}} |
|- | |- | ||
! [[Group (mathematics)|Group]] | ! [[Group (mathematics)|Group]] | ||
| − | | {{yes}} || {{yes}} || {{yes}} || {{yes}} || {{no}} | + | | {{yes|Required}} || {{yes|Required}} || {{yes|Required}} || {{yes|Required}} || {{no|Unneeded}} |
|- | |- | ||
! [[Abelian Group]] | ! [[Abelian Group]] | ||
| − | | {{yes}} || {{yes}} || {{yes}} || {{yes}} || {{yes}} | + | | {{yes|Required}} || {{yes|Required}} || {{yes|Required}} || {{yes|Required}} || {{yes|Required}} |
|- | |- | ||
| colspan="6" | <small>*[[Closure (mathematics)|Closure]], which is used in many sources, is an equivalent axiom to totality, though defined differently.</small> | | colspan="6" | <small>*[[Closure (mathematics)|Closure]], which is used in many sources, is an equivalent axiom to totality, though defined differently.</small> | ||
Revision as of 19:42, 5 July 2015
| Group-like structures. The entries say whether the property is required. | |||||
| Totality* | Associativity | Identity | Divisibility | Commutativity | |
|---|---|---|---|---|---|
| Semicategory | Unneeded | Required | Unneeded | Unneeded | Unneeded |
| Category | Unneeded | Required | Required | Unneeded | Unneeded |
| Groupoid | Unneeded | Required | Required | Required | Unneeded |
| Magma | Required | Unneeded | Unneeded | Unneeded | Unneeded |
| Quasigroup | Required | Unneeded | Unneeded | Required | Unneeded |
| Loop | Required | Unneeded | Required | Required | Unneeded |
| Semigroup | Required | Required | Unneeded | Unneeded | Unneeded |
| Monoid | Required | Required | Required | Unneeded | Unneeded |
| Group | Required | Required | Required | Required | Unneeded |
| Abelian Group | Required | Required | Required | Required | Required |
| *Closure, which is used in many sources, is an equivalent axiom to totality, though defined differently. | |||||
fr:Modèle:StructuresSemblablesGroupes pl:Szablon:Struktury grupopodobne