Difference between revisions of "Template:Group-like structures"
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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.) |
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| Line 21: | Line 21: | ||
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! [[Loop (algebra)|Loop]] | ! [[Loop (algebra)|Loop]] | ||
| − | | {{yes}} || {{no}} || {{yes}} || {{yes}} || {{no}} | + | | {{yes}} || {{no}} || {{yes}} || {{yes}}** || {{no}} |
|- | |- | ||
! [[Quasigroup]] | ! [[Quasigroup]] | ||
| Line 36: | Line 36: | ||
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| || 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