Difference between revisions of "Template:Group-like structures"
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imported>Gemisi m (Qualified the existence of inverses for loops: only left and right inverses exist.) |
imported>LokiClock (Defeats the purpose of the table, which is to classify the structures by their identities, visibly on inspection.) |
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| style="border-bottom: 2px solid #303060" colspan=6| '''Group-like structures''' | | style="border-bottom: 2px solid #303060" colspan=6| '''Group-like structures''' | ||
|- | |- | ||
| − | ! !! [[Total Function|Totality]]* !! [[Associativity]] !! [[Identity element|Identity]] !! [[ | + | ! !! [[Total Function|Totality]]* !! [[Associativity]] !! [[Identity element|Identity]] !! [[Quasigroup|Divisibility]] !! [[Commutativity]] |
|- | |- | ||
! [[Magma (algebra)|Magma]] | ! [[Magma (algebra)|Magma]] | ||
| Line 21: | Line 21: | ||
|- | |- | ||
! [[Loop (algebra)|Loop]] | ! [[Loop (algebra)|Loop]] | ||
| − | | {{yes}} || {{no}} || {{yes}} || {{yes}} | + | | {{yes}} || {{no}} || {{yes}} || {{yes}} || {{no}} |
|- | |- | ||
! [[Quasigroup]] | ! [[Quasigroup]] | ||
| − | | {{yes}} || {{no}} || {{no}} || {{ | + | | {{yes}} || {{no}} || {{no}} || {{yes}} || {{no}} |
|- | |- | ||
! [[Groupoid]] | ! [[Groupoid]] | ||
| 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> | ||
| − | |||
| − | |||
|} | |} | ||
<noinclude> | <noinclude> | ||
Revision as of 00:17, 11 March 2014
| Group-like structures | |||||
| Totality* | Associativity | Identity | Divisibility | 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 pl:Szablon:Struktury grupopodobne