Template:Group-like structures
Revision as of 21:08, 28 July 2020 by imported>Funandtrvl (refine category)
| Group-like structures | |||||
|---|---|---|---|---|---|
| Totalityα | Associativity | Identity | Invertibility | Commutativity | |
| Semigroupoid | Unneeded | Required | Unneeded | Unneeded | Unneeded |
| Small Category | Unneeded | Required | Required | Unneeded | Unneeded |
| Groupoid | Unneeded | Required | Required | Required | Unneeded |
| Magma | Required | Unneeded | Unneeded | Unneeded | Unneeded |
| Quasigroup | Required | Unneeded | Unneeded | Required | Unneeded |
| Unital Magma | Required | Unneeded | Required | Unneeded | Unneeded |
| Loop | Required | Unneeded | Required | Required | Unneeded |
| Semigroup | Required | Required | Unneeded | Unneeded | Unneeded |
| Inverse Semigroup | Required | Required | Unneeded | Required | Unneeded |
| Monoid | Required | Required | Required | Unneeded | Unneeded |
| Commutative monoid | Required | Required | Required | Unneeded | Required |
| Group | Required | Required | Required | Required | Unneeded |
| Abelian group | Required | Required | Required | Required | Required |
| ^α Closure, which is used in many sources, is an equivalent axiom to totality, though defined differently. | |||||