Участник:Dgerasimov/Algebra
Group-like structures
Taken from [1]
| Group-like structures | |||||
| Totality* | Associativity | Identity | Inverses | Commutativity | |
|---|---|---|---|---|---|
| Magma | Y | N | N | N | N |
| Semigroup | Y | Y | N | N | N |
| Monoid | Y | Y | Y | N | N |
| Commutative monoid | Y | Y | Y | N | Y |
| Group | Y | Y | Y | Y | N |
| Abelian Group | Y | Y | Y | Y | Y |
| Loop | Y | N | Y | Y** | N |
| Quasigroup | Y | N | N | N | N |
| Groupoid | N | Y | Y | Y | N |
| Category | N | Y | Y | N | N |
| Semicategory | N | Y | N | N | N |
| *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. | |||||
Ring-like structures
| Ring-like structures | |||||||
| Magma* | Semigroup | Commutative semigroup | Monoid | Commutative monoid | Group | Abelian Group | |
|---|---|---|---|---|---|---|---|
| Magma | Ringoid | ? | N | N | ? | N | N |
| Semigroup | Y | Semiring | ? | N | ? | N | N |
| Commutative semigroup | ? | Additive semiring, semiring | ? | ? | ? | ? | ? |
| Monoid | Y | Rig | N | N | ? | N | N |
| Commutative monoid | ? | Rig | ? | ? | ? | ? | ? |
| Group | Y | ? | N | N | ? | N | N |
| Abelian Group | ? | Ring | ? | Unital ring | Commutative ring | Division ring (algebra) | Field |
