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

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{| class="infobox" style="float:right; text-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="border-bottom: 2px solid #303060; text-align:center;" colspan=6 | '''Group-like structures. The entries say whether the property is ''required''.'''
 
|-
 
|-
 
! !! [[Total Function|Totality]]* !! [[Associativity]] !! [[Identity element|Identity]] !! [[Inverse element|Divisibility]] !! [[Commutativity]]
 
! !! [[Total Function|Totality]]* !! [[Associativity]] !! [[Identity element|Identity]] !! [[Inverse element|Divisibility]] !! [[Commutativity]]
 
|-
 
|-
 
! [[Semicategory]]
 
! [[Semicategory]]
| {{no}} || {{yes}} || {{no}} || {{no}} || {{no}}
+
| {{no|Unneeded}} || {{yes|Required}} || {{no|Unneeded}} || {{no|Unneeded}} || {{no|Unneeded}}
 
|-
 
|-
 
! [[Category (mathematics)|Category]]
 
! [[Category (mathematics)|Category]]
| {{no}} || {{yes}} || {{yes}} || {{no}} || {{no}}
+
| {{no|Unneeded}} || {{yes|Required}} || {{yes|Required}} || {{no|Unneeded}} || {{no|Unneeded}}
 
|-
 
|-
 
! [[Groupoid]]
 
! [[Groupoid]]
| {{no}} || {{yes}} || {{yes}} || {{yes}} || {{no}}
+
| {{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]]
 
! [[Quasigroup]]
| {{yes}} || {{no}} || {{no}} || {{yes}} || {{no}}
+
| {{yes|Required}} || {{no|Unneeded}} || {{no|Unneeded}} || {{yes|Required}} || {{no|Unneeded}}
 
|-
 
|-
 
! [[Loop (algebra)|Loop]]
 
! [[Loop (algebra)|Loop]]
| {{yes}} || {{no}} || {{yes}} || {{yes}} || {{no}}
+
| {{yes|Required}} || {{no|Unneeded}} || {{yes|Required}} || {{yes|Required}} || {{no|Unneeded}}
 
|-
 
|-
 
! [[Semigroup]]
 
! [[Semigroup]]
| {{yes}} || {{yes}} || {{no}} || {{no}}  || {{no}}
+
| {{yes|Required}} || {{yes|Required}} || {{no|Unneeded}} || {{no|Unneeded}}  || {{no|Unneeded}}
 
|-
 
|-
 
! [[Monoid]]
 
! [[Monoid]]
| {{yes}} || {{yes}} || {{yes}} || {{no}} || {{no}}
+
| {{yes|Required}} || {{yes|Required}} || {{yes|Required}} || {{no|Unneeded}} || {{no|Unneeded}}
 
|-
 
|-
 
! [[Group (mathematics)|Group]]
 
! [[Group (mathematics)|Group]]
| {{yes}} || {{yes}} || {{yes}} || {{yes}} || {{no}}
+
| {{yes|Required}} || {{yes|Required}} || {{yes|Required}} || {{yes|Required}} || {{no|Unneeded}}
 
|-
 
|-
 
! [[Abelian Group]]
 
! [[Abelian Group]]
| {{yes}} || {{yes}} || {{yes}} || {{yes}} || {{yes}}
+
| {{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>
 
| colspan="6" | <small>*[[Closure (mathematics)|Closure]], which is used in many sources, is an equivalent axiom to totality, though defined differently.</small>

Revision as of 19:42, 5 July 2015

Group-like structures. The entries say whether the property is required.
Totality* 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
*Closure, which is used in many sources, is an equivalent axiom to totality, though defined differently.

fr:Modèle:StructuresSemblablesGroupes pl:Szablon:Struktury grupopodobne