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

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(Shrank footnote so the table as a whole isn't as wide.)
(Reordered rows so later rows add properties to earlier rows. Exceptions have a "reflection pattern" that is hopefully mnemonic. WP:BB)
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|-
 
|-
 
! !! [[Total Function|Totality]]* !! [[Associativity]] !! [[Identity element|Identity]] !! [[Inverse element|Divisibility]] !! [[Commutativity]]
 
! !! [[Total Function|Totality]]* !! [[Associativity]] !! [[Identity element|Identity]] !! [[Inverse element|Divisibility]] !! [[Commutativity]]
 +
|-
 +
! [[Semicategory]]
 +
| {{no}} || {{yes}} || {{no}} || {{no}} || {{no}}
 +
|-
 +
! [[Category (mathematics)|Category]]
 +
| {{no}} || {{yes}} || {{yes}} || {{no}} || {{no}}
 +
|-
 +
! [[Groupoid]]
 +
| {{no}} || {{yes}} || {{yes}} || {{yes}} || {{no}}
 
|-
 
|-
 
! [[Magma (algebra)|Magma]]
 
! [[Magma (algebra)|Magma]]
 
| {{yes}} || {{no}} || {{no}} || {{no}} || {{no}}
 
| {{yes}} || {{no}} || {{no}} || {{no}} || {{no}}
 +
|-
 +
! [[Quasigroup]]
 +
| {{yes}} || {{no}} || {{no}} || {{yes}} || {{no}}
 +
|-
 +
! [[Loop (algebra)|Loop]]
 +
| {{yes}} || {{no}} || {{yes}} || {{yes}} || {{no}}
 
|-
 
|-
 
! [[Semigroup]]
 
! [[Semigroup]]
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! [[Abelian Group]]
 
! [[Abelian Group]]
 
| {{yes}} || {{yes}} || {{yes}} || {{yes}} || {{yes}}
 
| {{yes}} || {{yes}} || {{yes}} || {{yes}} || {{yes}}
|-
 
! [[Loop (algebra)|Loop]]
 
| {{yes}} || {{no}} || {{yes}} || {{yes}} || {{no}}
 
|-
 
! [[Quasigroup]]
 
| {{yes}} || {{no}} || {{no}} || {{yes}} || {{no}}
 
|-
 
! [[Groupoid]]
 
| {{no}} || {{yes}} || {{yes}} || {{yes}} || {{no}}
 
|-
 
! [[Category (mathematics)|Category]]
 
| {{no}} || {{yes}} || {{yes}} || {{no}} || {{no}}
 
|-
 
! [[Semicategory]]
 
| {{no}} || {{yes}} || {{no}} || {{no}} || {{no}}
 
 
|-
 
|-
 
| 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 15:33, 12 November 2014

Group-like structures. The entries say whether the property is required.
Totality* Associativity Identity Divisibility Commutativity
Semicategory No Yes No No No
Category No Yes Yes No No
Groupoid No Yes Yes Yes No
Magma Yes No No No No
Quasigroup Yes No No Yes No
Loop Yes No Yes Yes No
Semigroup Yes Yes No No No
Monoid Yes Yes Yes No No
Group Yes Yes Yes Yes No
Abelian Group Yes Yes Yes Yes Yes
*Closure, which is used in many sources, is an equivalent axiom to totality, though defined differently.

fr:Modèle:StructuresSemblablesGroupes pl:Szablon:Struktury grupopodobne