Algebraic Structure
An algebraic structure consists of:
- a nonempty set $A$
- a collection of operations on $A$ that satisfy a finite set of axioms.
Group-like Structures
| Total | Associative | Identity | Divisible | Commutative | |
|---|---|---|---|---|---|
| Partial Magma | $\quad$ | $\quad$ | $\quad$ | $\quad$ | $\quad$ |
| Semigroupoid | $\quad$ | $\checkmark$ | $\quad$ | $\quad$ | $\quad$ |
| Category | $\quad$ | $\checkmark$ | $\checkmark$ | $\quad$ | $\quad$ |
| Groupoid | $\quad$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\quad$ |
| Commutative Groupoid | $\quad$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ |
| Magma | $\checkmark$ | $\quad$ | $\quad$ | $\quad$ | $\quad$ |
| Commutative Magma | $\checkmark$ | $\quad$ | $\quad$ | $\quad$ | $\checkmark$ |
| Quasigroup | $\checkmark$ | $\quad$ | $\quad$ | $\checkmark$ | $\quad$ |
| Commutative Quasigroup | $\checkmark$ | $\quad$ | $\quad$ | $\checkmark$ | $\checkmark$ |
| Unital Magma | $\checkmark$ | $\quad$ | $\checkmark$ | $\quad$ | $\quad$ |
| Commutative unital magma | $\checkmark$ | $\quad$ | $\checkmark$ | $\quad$ | $\checkmark$ |
| Loop | $\checkmark$ | $\quad$ | $\checkmark$ | $\checkmark$ | $\quad$ |
| Commutative Loop | $\checkmark$ | $\quad$ | $\checkmark$ | $\checkmark$ | $\checkmark$ |
| Semigroup | $\checkmark$ | $\checkmark$ | $\quad$ | $\quad$ | $\quad$ |
| Commutative Semigroup | $\checkmark$ | $\checkmark$ | $\quad$ | $\quad$ | $\checkmark$ |
| Associative Quasigroup | $\checkmark$ | $\checkmark$ | $\quad$ | $\checkmark$ | $\quad$ |
| Commutative-and-associative Quasigroup | $\checkmark$ | $\checkmark$ | $\quad$ | $\checkmark$ | $\checkmark$ |
| Monoid | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\quad$ | $\quad$ |
| Commutative Monoid | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\quad$ | $\checkmark$ |
| Group | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\quad$ |
| Abelian Group | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ |