Difference between revisions of "Template:Group-like structures"
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imported>Quondum (Moving badly placed line break in caption) |
imported>R160K (Added a footnote explaining the equivalence of "closure" and "totality" - there have been various edits back and forth between "closure" and "totality", although I hope nobody will object to a footnote) |
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{| align="right" class="infobox" style="text-align:center" | {| align="right" class="infobox" style="text-align:center" | ||
|- align="center" | |- align="center" | ||
| − | | style="border-bottom: 2px solid #303060" colspan= | + | | style="border-bottom: 2px solid #303060" colspan=5| '''Group-like structures''' |
|- | |- | ||
| − | ! !! [[Total Function|Totality]] !! [[ | + | ! !! [[Total Function|Totality]]* !! [[Associativity]] !! [[Identity element|Identity]] !! [[Inverse element|Inverses]] !! [[Commutativity]] |
|- | |- | ||
! [[Magma (algebra)|Magma]] | ! [[Magma (algebra)|Magma]] | ||
| {{yes}} || {{no}} || {{no}} || {{no}} || {{no}} | | {{yes}} || {{no}} || {{no}} || {{no}} || {{no}} | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
|- | |- | ||
! [[Semigroup]] | ! [[Semigroup]] | ||
| − | | {{yes}} || {{ | + | | {{yes}} || {{yes}} || {{no}} || {{no}} || {{no}} |
|- | |- | ||
! [[Monoid]] | ! [[Monoid]] | ||
| Line 28: | Line 19: | ||
! [[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]] | ! [[Groupoid]] | ||
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|- | |- | ||
! [[Semicategory]] | ! [[Semicategory]] | ||
| − | | {{no}} || {{ | + | | {{no}} || {{yes}} || {{no}} || {{no}} || {{no}} |
| − | |||
| − | |||
|} | |} | ||
| + | <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> | ||
<noinclude> | <noinclude> | ||
[[Category:Mathematics templates|{{PAGENAME}}]] | [[Category:Mathematics templates|{{PAGENAME}}]] | ||
| − | |||
| − | |||
[[fr:Modèle:StructuresSemblablesGroupes]] | [[fr:Modèle:StructuresSemblablesGroupes]] | ||
[[ja:Template:Group-like structures]] | [[ja:Template:Group-like structures]] | ||
Revision as of 16:27, 4 November 2012
| 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 | Yes | 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.
fr:Modèle:StructuresSemblablesGroupes ja:Template:Group-like structures pl:Szablon:Struktury grupopodobne sv:Mall:Gruppliknande strukturer zh:Template:類群結構