Definition:Properties of Algebraic Structures of One Operation
Jump to navigation
Jump to search
Definition
The purpose of this page is to gather into one place the various types of algebraic structure of one binary operation and classify them according to the properties they hold.
Closure | Associativity | Identity | Inverses | Commutativity | Latin Square Property | |
---|---|---|---|---|---|---|
Magma | $\checkmark$ | |||||
Quasigroup | $\checkmark$ | $\checkmark$ | ||||
Algebra Loop | $\checkmark$ | $\checkmark$ | $\checkmark$ | |||
Semigroup | $\checkmark$ | $\checkmark$ | ||||
Commutative Semigroup | $\checkmark$ | $\checkmark$ | $\checkmark$ | |||
Monoid | $\checkmark$ | $\checkmark$ | $\checkmark$ | |||
Commutative Monoid | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | ||
Group | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $(\checkmark)$: see here | |
Abelian Group | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $(\checkmark)$: see here |
A checkmark in brackets: $(\checkmark)$ denotes that the property indicated can be derived from the others.