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.
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- Note that both divisibility and invertibility
- imply the cancellation property.
Read more about this topic: Magma (algebra)
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