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}} | ||
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| 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