Difference between revisions of "Template:Group-like structures"
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! style="background:#cee0f2; padding:0.2em 0.4em 0.2em; text-align:center; font-size:130%; line-height:1.2em;" colspan=6 | Group-like structures | ! style="background:#cee0f2; padding:0.2em 0.4em 0.2em; text-align:center; font-size:130%; line-height:1.2em;" colspan=6 | Group-like structures | ||
|- | |- | ||
| − | ! !! [[Total Function|Totality]] | + | ! !! [[Total Function|Totality]]{{ref|Alpha|α}} !! [[Associativity]] !! [[Identity element|Identity]] !! [[Inverse element|Divisibility]] !! [[Commutativity]] |
|- | |- | ||
! [[Semicategory]] | ! [[Semicategory]] | ||
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| {{yes|Required}} || {{yes|Required}} || {{yes|Required}} || {{yes|Required}} || {{yes|Required}} | | {{yes|Required}} || {{yes|Required}} || {{yes|Required}} || {{yes|Required}} || {{yes|Required}} | ||
|- | |- | ||
| − | | colspan="6" | <small> | + | | colspan="6" | <small>{{note|Alpha|α}}[[Closure (mathematics)|Closure]], which is used in many sources, is an equivalent axiom to totality, though defined differently.</small> |
|} | |} | ||
<noinclude> | <noinclude> | ||
Revision as of 10:32, 6 July 2015
| Group-like structures | |||||
|---|---|---|---|---|---|
| Totalityα | 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 |
| ^α Closure, which is used in many sources, is an equivalent axiom to totality, though defined differently. | |||||
fr:Modèle:StructuresSemblablesGroupes pl:Szablon:Struktury grupopodobne