Definition:Properties of Algebraic Structures of One Operation

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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.


Also see