Difference between revisions of "Template:Group-like structures"
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(Shrank footnote so the table as a whole isn't as wide.) |
imported>Funandtrvl (refine category) |
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− | {| | + | {| 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> | + | | 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 | ||
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</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. |