Algebraic structure

Roughly speaking, an algebraic structure is a set \(X\), known as the underlying set, paired with a collection of operations that obey a given set of laws. For example, a group is a set paired with a single binary operation that satisfies the four group axioms, and a ring is a set paired with two binary operations that satisfy the ten ring axioms.

In fact, algebraic structures can have more than one underlying set. Most have only one (including monoids, groups, rings, fields, lattices, and arithmetics), and differ in how their associated operations work. More complex algebraic structures (such as algebras, modules, and vector spaces) have two underlying sets. For example, vector spaces are defined using both an underlying field of scalars and an underlying commutative group of vectors.

For a map of algebraic structures and how they relate to each other, see the tree of algebraic structures.


  • Group

    The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.

  • Ring
  • Abelian group

    A group where the operation commutes. Named after Niels Henrik Abel.

  • Monoid
  • Algebraic structure tree

    When is a monoid a semilattice? What’s the difference between a semigroup and a groupoid? Find out here!


  • Abstract algebra

    The study of groups, fields, vector spaces, arithmetics, algebras, and more.