In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties. The notion of algebraic structure has been formalized in universal algebra.
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;
- 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.
- From magma to group, via two alternative paths. Key:
- M = magma, d = divisibility, a = associativity,
- Q = quasigroup, S = semigroup, e = identity.
- L = loop, i = invertibility, N = monoid, G = group
- Note that both divisibility and invertibility imply
- the existence of the cancellation property.