Algebraic structure

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:
Simple structures
(Although some mathematicians would not count the following as algebraic structures, we include them for completeness)
 Set: a set can itself be thought of as a degenerate algebraic structure, one that has zero operations defined on it
 Pointed set: a set S with a distinguished element s of S
 Unary system: a set S with a unary operation, i.e. a function S → S
 Pointed unary system: a unary system with a distinguished element (such objects occur in discussions of the Peano axioms)
Grouplike 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
 Archimedean group: a linearly ordered group for which the Archimedean property holds
Ringlike 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 nonzero 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 nonassociative 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 nonalgebraic 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 categorytheoretic sense. Similarly, the category of topological groups (with continuous group homomorphisms as morphisms) is a category of topological spaces with extra structure.
See also: signature (universal algebra)de:Algebraische Struktur fr:Structure algébrique it:Struttura algebrica he:מבנה אלגברי ja:代数的構造 nl:Algebraïsche structuur pt:Estrutura Algébrica ru:Алгебраическая система sv:Algebraisk struktur zh:代数结构