Algebraic structure - Definition


Algebraic structure - Definition

In abstract algebra, an algebraic structure consists of a set together with a collection of operations or relations defined on it which satisfy certain axioms. When there are no ambiguities, we usually identify the set with the algebraic structure. For example, a group (G,*) is usually referred simply as a group G. If there are only relations and no operations, we speak of a relational structure.

Depending on the operations, relations and axioms, the algebraic structures get their names. The following is a partial list of algebraic structures:

Set: although some mathematicians would not count it, a set can itself be thought of as a degenerate algebraic structure, one that has zero operations defined on it

Group-like structures

Magma or groupoid: a set with a single binary operation
Quasigroup: a magma in which division is always possible
Loop: a quasigroup with an identity element
Semigroup: an associative magma
Monoid: a semigroup with an identity element
Group: a monoid in which every element has an inverse, or equivalently, an associative loop
Abelian group: a commutative group

Ring-like structures

Semiring: similar to a Ring, but without additive inverses
Ring: a set with an abelian group operation as addition, together with a monoid operation as multiplication, satisfying distributivity
Commutative ring: a ring whose multiplication is commutative
Division ring: a ring with 0 ≠ 1 in which each non-zero element has an inverse
Field: a commutative division ring
Kleene algebra: an idempotent semiring with additional unary operator (the Kleene star); these are modeled on regular expressions

Modules

Module over a given ring R: a set with an abelian group operation as addition, together with an additive unary operation of scalar multiplication for every element of R, with an associativity condition linking scalar multiplication to multiplication in R
Vector space: a module over a field

Algebras

Algebra: a module or vector space together with a bilinear operation as multiplication
Associative algebra: an algebra whose multiplication is associative
Commutative algebra: an associative algebra whose multiplication is commutative
Graded algebra: an algebra with a "grading"
Lie algebra: a non-associative algebra important in geometry
Clifford algebra: an associative algebra determined by quadratic form on a vector space

Lattices

Lattice: a set with two commutative, associative, idempotent operations satisfying the absorption law
Boolean algebra: a bounded, distributive, complemented lattice

Those statements that apply to all algebraic structures collectively are investigated in the branch of mathematics known as universal algebra.

Algebraic structures can also be defined on sets with additional non-algebraic structures, such as topological spaces. For example, a topological group is a topological space with a group structure such that the operations of multiplication and taking inverses are continuous; a topological group has both a topological and an algebraic structure. Other common examples are topological vector spaces and Lie groups.

Every algebraic structure has its own notion of homomorphism, a function that is compatible with the given operation(s). In this way, every algebraic structure defines a category. For example, the category of groups has all groups as objects and all group homomorphisms as morphisms. This category, being a concrete category, may be regarded as a category of sets with extra structure in the category-theoretic sense. Similarly, the category of topological groups (with continuous group homomorphisms as morphisms) is a category of topological spaces with extra structure.