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

From blackwiki
Jump to navigation Jump to search
imported>Gemisi
m (Inverses are not defined in general quasigroups; there is not even an identity. But unlike a general magma, left- and right-multiplication maps are bijective)
imported>Gemisi
m (Qualified the existence of inverses for loops: only left and right inverses exist.)
Line 21: Line 21:
 
|-
 
|-
 
! [[Loop (algebra)|Loop]]
 
! [[Loop (algebra)|Loop]]
| {{yes}} || {{no}} || {{yes}} || {{yes}} || {{no}}
+
| {{yes}} || {{no}} || {{yes}} || {{yes}}** || {{no}}
 
|-
 
|-
 
! [[Quasigroup]]
 
! [[Quasigroup]]
Line 36: Line 36:
 
|-
 
|-
 
| || colspan="5" | <small>*[[Closure (mathematics)|Closure]], which is used in many sources to define group-like structures, is an equivalent axiom to totality, though defined differently.</small>
 
| || colspan="5" | <small>*[[Closure (mathematics)|Closure]], which is used in many sources to define group-like structures, is an equivalent axiom to totality, though defined differently.</small>
 +
|-
 +
| || colspan="5" | <small>**Each element of a [[Loop (algebra)|loop]] has a left and right inverse, but these need not coincide.</small>
 
|}
 
|}
 
<noinclude>
 
<noinclude>

Revision as of 16:56, 19 August 2013

Group-like structures
Totality* Associativity Identity Inverses Commutativity
Magma Yes No No No 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
Loop Yes No Yes Yes** No
Quasigroup Yes No No No No
Groupoid No Yes Yes Yes No
Category No Yes Yes No No
Semicategory No Yes No No No
*Closure, which is used in many sources to define group-like structures, is an equivalent axiom to totality, though defined differently.
**Each element of a loop has a left and right inverse, but these need not coincide.

fr:Modèle:StructuresSemblablesGroupes pl:Szablon:Struktury grupopodobne