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.

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