Template:Group-like structures
Revision as of 16:27, 4 November 2012 by 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)
| 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.
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