Magma (algebra) - Types of Magmas

Types of Magmas

Magmas are not often studied as such; instead there are several different kinds of magmas, depending on what axioms one might require of the operation. Commonly studied types of magmas include

• quasigroups—nonempty magmas where division is always possible;
• loops—quasigroups with identity elements;
• semigroups—magmas where the operation is associative;
• semilattices—semigroups where the operation is commutative and idempotent;
• monoids—semigroups with identity elements;
• groups—monoids with inverse elements, or equivalently, associative loops or associative quasigroups;
• abelian groups—groups where the operation is commutative.

Note that both divisibility and invertibility

imply the cancellation property.