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

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(Shrank footnote so the table as a whole isn't as wide.)
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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"
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|-
| style="border-bottom: 2px solid #303060" colspan=6| '''Group-like structures. The entries say whether the property is ''required''.'''
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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
 
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|-
! !! [[Total Function|Totality]]* !! [[Associativity]] !! [[Identity element|Identity]] !! [[Inverse element|Divisibility]] !! [[Commutativity]]
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! !! style="text-align:center" |[[Total function|Totality]]{{ref|Alpha|α}}
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!! style="text-align:center" |[[Associativity]]  
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!! style="text-align:center" | [[Identity element|Identity]]  
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!! style="text-align:center" | [[Inverse element|Invertibility]]  
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!! style="text-align:center" | [[Commutativity]]
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|-
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! style = "display:block" | [[Semigroupoid]]
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| {{no|Unneeded}} || {{yes|Required}} || {{no|Unneeded}} || {{no|Unneeded}} || {{no|Unneeded}}
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|-
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! [[Category (mathematics)|Small Category]]
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| {{no|Unneeded}} || {{yes|Required}} || {{yes|Required}} || {{no|Unneeded}} || {{no|Unneeded}}
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|-
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! [[Groupoid]]
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| {{no|Unneeded}} || {{yes|Required}} || {{yes|Required}} || {{yes|Required}} || {{no|Unneeded}}
 
|-
 
|-
 
! [[Magma (algebra)|Magma]]
 
! [[Magma (algebra)|Magma]]
| {{yes}} || {{no}} || {{no}} || {{no}} || {{no}}
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| {{yes|Required}} || {{no|Unneeded}} || {{no|Unneeded}} || {{no|Unneeded}} || {{no|Unneeded}}
 
|-
 
|-
! [[Semigroup]]
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! [[Quasigroup]]
| {{yes}} || {{yes}} || {{no}} || {{no}} || {{no}}
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| {{yes|Required}} || {{no|Unneeded}} || {{no|Unneeded}} || {{yes|Required}} || {{no|Unneeded}}
 
|-
 
|-
! [[Monoid]]
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! [[Unital Magma]]
| {{yes}} || {{yes}} || {{yes}} || {{no}} || {{no}}
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| {{yes|Required}} || {{no|Unneeded}} ||{{yes|Required}} || {{no|Unneeded}} || {{no|Unneeded}}
 
|-
 
|-
! [[Group (mathematics)|Group]]
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! [[Loop (algebra)|Loop]]
| {{yes}} || {{yes}} || {{yes}} || {{yes}} || {{no}}
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| {{yes|Required}} || {{no|Unneeded}} || {{yes|Required}} || {{yes|Required}} || {{no|Unneeded}}
 
|-
 
|-
! [[Abelian Group]]
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! [[Semigroup]]
| {{yes}} || {{yes}} || {{yes}} || {{yes}} || {{yes}}
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| {{yes|Required}} || {{yes|Required}} || {{no|Unneeded}} || {{no|Unneeded}} || {{no|Unneeded}}
 
|-
 
|-
! [[Loop (algebra)|Loop]]
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! [[Inverse_semigroup|Inverse Semigroup]]
| {{yes}} || {{no}} || {{yes}} || {{yes}} || {{no}}
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| {{yes|Required}} || {{yes|Required}} || {{no|Unneeded}} || {{yes|Required}} || {{no|Unneeded}}
 
|-
 
|-
! [[Quasigroup]]
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! [[Monoid]]
| {{yes}} || {{no}} || {{no}} || {{yes}} || {{no}}
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| {{yes|Required}} || {{yes|Required}} || {{yes|Required}} || {{no|Unneeded}} || {{no|Unneeded}}
 
|-
 
|-
! [[Groupoid]]
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! [[Commutative monoid]]
| {{no}} || {{yes}} || {{yes}} || {{yes}} || {{no}}
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| {{yes|Required}} || {{yes|Required}} || {{yes|Required}} || {{no|Unneeded}} || {{yes|Required}}
 
|-
 
|-
! [[Category (mathematics)|Category]]
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! [[Group (mathematics)|Group]]
| {{no}} || {{yes}} || {{yes}} || {{no}} || {{no}}
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| {{yes|Required}} || {{yes|Required}} || {{yes|Required}} || {{yes|Required}} || {{no|Unneeded}}
 
|-
 
|-
! [[Semicategory]]
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! [[Abelian group]]
| {{no}} || {{yes}} || {{no}} || {{no}} || {{no}}
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| {{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>
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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>
|}
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|}<noinclude>
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[[Category:Mathematics sidebar templates]]
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[[fr:Modèle:StructuresSemblablesGroupes]]
 
[[pl:Szablon:Struktury grupopodobne]]
 
 
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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.