Difference between revisions of "Template:Group-like structures"

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imported>KamikazeBot
m (r2.7.3) (Robot: Adding af:Sjabloon:Groepagtige strukture)
imported>Funandtrvl
(refine category)
 
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{| align="right" class="infobox" style="text-align:center"
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{| style="margin:0.5em 0 1em 35px;float:right!important;font-size:90%"
|- align="center"
+
|-
| style="border-bottom: 2px solid #303060" colspan=6| '''Requirements of 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]] !! [[Identity element|Identity]] !! [[Associativity]] !! [[Inverse element|Inverses]] !! [[Commutativity]]
+
! !! 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}}
 
|-
 
|-
! [[Unital magma|Unital]]
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! [[Quasigroup]]
| {{yes}} || {{yes}} || {{no}} || {{no}} || {{no}}
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| {{yes|Required}} || {{no|Unneeded}} || {{no|Unneeded}} || {{yes|Required}} || {{no|Unneeded}}
 
|-
 
|-
! [[Quasigroup]]
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! [[Unital Magma]]
| {{yes}} || {{no}} || {{no}} || {{yes}} || {{no}}
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| {{yes|Required}} || {{no|Unneeded}} ||{{yes|Required}} || {{no|Unneeded}} || {{no|Unneeded}}
 
|-
 
|-
 
! [[Loop (algebra)|Loop]]
 
! [[Loop (algebra)|Loop]]
| {{yes}} || {{yes}} || {{no}} || {{yes}} || {{no}}
+
| {{yes|Required}} || {{no|Unneeded}} || {{yes|Required}} || {{yes|Required}} || {{no|Unneeded}}
 
|-
 
|-
 
! [[Semigroup]]
 
! [[Semigroup]]
| {{yes}} || {{no}} || {{yes}} || {{no}}  || {{no}}
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| {{yes|Required}} || {{yes|Required}} || {{no|Unneeded}} || {{no|Unneeded}} || {{no|Unneeded}}
 +
|-
 +
! [[Inverse_semigroup|Inverse Semigroup]]
 +
| {{yes|Required}} || {{yes|Required}} || {{no|Unneeded}} || {{yes|Required}}  || {{no|Unneeded}}
 
|-
 
|-
 
! [[Monoid]]
 
! [[Monoid]]
| {{yes}} || {{yes}} || {{yes}} || {{no}} || {{no}}
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| {{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]]
 
! [[Group (mathematics)|Group]]
| {{yes}} || {{yes}} || {{yes}} || {{yes}} || {{no}}
+
| {{yes|Required}} || {{yes|Required}} || {{yes|Required}} || {{yes|Required}} || {{no|Unneeded}}
|-
 
! [[Abelian Group]]
 
| {{yes}} || {{yes}} || {{yes}} || {{yes}} || {{yes}}
 
|-
 
! [[Groupoid]]
 
| {{no}} || {{yes}} || {{yes}} || {{yes}} || {{no}}
 
 
|-
 
|-
! [[Category (mathematics)|Category]]
+
! [[Abelian group]]
| {{no}} || {{yes}} || {{yes}} || {{no}} || {{no}}
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| {{yes|Required}} || {{yes|Required}} || {{yes|Required}} || {{yes|Required}} || {{yes|Required}}
 
|-
 
|-
! [[Semicategory]]
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| 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>
| {{no}} || {{no}} || {{yes}} || {{no}} || {{no}}
+
|}<noinclude>
|- align="center"
+
<!-- Categories below this line; interwikis at Wikidata -->
| style="border-top: 2px solid #303060" colspan=6| '''''Note:'''''  ''A quasigroup with associativity (equivalently, a semigroup with inverses) <br /> already has an [[identity element]], and is therefore a group.'' 
+
[[Category:Mathematics sidebar templates]]
|}
 
<noinclude>
 
 
 
[[Category:Mathematics templates|{{PAGENAME}}]]
 
 
 
[[af:Sjabloon:Groepagtige strukture]]
 
[[cs:Šablona:Grupové struktury]]
 
[[fr:Modèle:StructuresSemblablesGroupes]]
 
[[ja:Template:Group-like structures]]
 
[[pl:Szablon:Struktury grupopodobne]]
 
[[sv:Mall:Gruppliknande strukturer]]
 
[[zh:Template:類群結構]]
 
 
</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
Closure, which is used in many sources, is an equivalent axiom to totality, though defined differently.