Difference between revisions of "Template:Group-like structures"
Jump to navigation
Jump to search
imported>Eric Kvaalen (The entries say whether the property is required.) |
imported>Funandtrvl (refine category) |
||
| (26 intermediate revisions by 17 users not shown) | |||
| Line 1: | Line 1: | ||
| − | {| | + | {| style="margin:0.5em 0 1em 35px;float:right!important;font-size:90%" |
| − | + | |- | |
| − | + | ! 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 | + | ! !! style="text-align:center" |[[Total function|Totality]]{{ref|Alpha|α}} |
| + | !! style="text-align:center" |[[Associativity]] | ||
| + | !! style="text-align:center" | [[Identity element|Identity]] | ||
| + | !! style="text-align:center" | [[Inverse element|Invertibility]] | ||
| + | !! style="text-align:center" | [[Commutativity]] | ||
| + | |- | ||
| + | ! style = "display:block" | [[Semigroupoid]] | ||
| + | | {{no|Unneeded}} || {{yes|Required}} || {{no|Unneeded}} || {{no|Unneeded}} || {{no|Unneeded}} | ||
| + | |- | ||
| + | ! [[Category (mathematics)|Small Category]] | ||
| + | | {{no|Unneeded}} || {{yes|Required}} || {{yes|Required}} || {{no|Unneeded}} || {{no|Unneeded}} | ||
| + | |- | ||
| + | ! [[Groupoid]] | ||
| + | | {{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]] |
| − | | {{yes}} || {{ | + | | {{yes|Required}} || {{no|Unneeded}} || {{no|Unneeded}} || {{yes|Required}} || {{no|Unneeded}} |
|- | |- | ||
| − | ! [[ | + | ! [[Unital Magma]] |
| − | | {{yes}} || {{ | + | | {{yes|Required}} || {{no|Unneeded}} ||{{yes|Required}} || {{no|Unneeded}} || {{no|Unneeded}} |
|- | |- | ||
| − | ! [[ | + | ! [[Loop (algebra)|Loop]] |
| − | | {{yes}} || {{ | + | | {{yes|Required}} || {{no|Unneeded}} || {{yes|Required}} || {{yes|Required}} || {{no|Unneeded}} |
|- | |- | ||
| − | ! [[ | + | ! [[Semigroup]] |
| − | | {{yes}} || {{yes}} || {{ | + | | {{yes|Required}} || {{yes|Required}} || {{no|Unneeded}} || {{no|Unneeded}} || {{no|Unneeded}} |
|- | |- | ||
| − | ! [[ | + | ! [[Inverse_semigroup|Inverse Semigroup]] |
| − | | {{yes}} || {{ | + | | {{yes|Required}} || {{yes|Required}} || {{no|Unneeded}} || {{yes|Required}} || {{no|Unneeded}} |
|- | |- | ||
| − | ! [[ | + | ! [[Monoid]] |
| − | | {{yes}} || {{ | + | | {{yes|Required}} || {{yes|Required}} || {{yes|Required}} || {{no|Unneeded}} || {{no|Unneeded}} |
|- | |- | ||
| − | ! [[ | + | ! [[Commutative monoid]] |
| − | | {{ | + | | {{yes|Required}} || {{yes|Required}} || {{yes|Required}} || {{no|Unneeded}} || {{yes|Required}} |
|- | |- | ||
| − | ! [[ | + | ! [[Group (mathematics)|Group]] |
| − | | {{ | + | | {{yes|Required}} || {{yes|Required}} || {{yes|Required}} || {{yes|Required}} || {{no|Unneeded}} |
|- | |- | ||
| − | ! [[ | + | ! [[Abelian group]] |
| − | | {{ | + | | {{yes|Required}} || {{yes|Required}} || {{yes|Required}} || {{yes|Required}} || {{yes|Required}} |
|- | |- | ||
| − | + | | 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> | + | <!-- Categories below this line; interwikis at Wikidata --> |
| − | + | [[Category:Mathematics sidebar templates]] | |
| − | [[Category:Mathematics templates | ||
| − | |||
| − | |||
| − | |||
</noinclude> | </noinclude> | ||
Latest revision as of 21:08, 28 July 2020
| Group-like structures | |||||
|---|---|---|---|---|---|
| Totalityα | Associativity | Identity | Invertibility | Commutativity | |
| Semigroupoid | Unneeded | Required | Unneeded | Unneeded | Unneeded |
| Small Category | Unneeded | Required | Required | Unneeded | Unneeded |
| Groupoid | Unneeded | Required | Required | Required | Unneeded |
| Magma | Required | Unneeded | Unneeded | Unneeded | Unneeded |
| Quasigroup | Required | Unneeded | Unneeded | Required | Unneeded |
| Unital Magma | Required | Unneeded | Required | Unneeded | Unneeded |
| Loop | Required | Unneeded | Required | Required | Unneeded |
| Semigroup | Required | Required | Unneeded | Unneeded | Unneeded |
| Inverse Semigroup | Required | Required | Unneeded | Required | Unneeded |
| Monoid | Required | Required | Required | Unneeded | Unneeded |
| Commutative monoid | Required | Required | Required | Unneeded | Required |
| 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. | |||||