Algebraic structures
From Exampleproblems
In higher mathematics, "algebraic structure" is a loosely-defined phrase referring to the mathematical objects traditionally studied in the field of abstract algebra: sets with operations.
The word "structure" can refer to a specific mathematical object or an even more abstract concept. For example, the monster group simultaneously is an algebraic structure, and it has an algebraic structure: the structure shared by all groups. This article uses both senses of the phrase.
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In the sense of universal algebra
In universal algebra, one studies algebraic structures consisting of a set and a collection of operations defined on the set which are required to satisfy certain identities.
Simple structures
- Set: a set is 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)
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
- 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
- Lie algebra: a non-associative algebra important in geometry
Lattices
- Lattice: a set with two commutative, associative, idempotent operations satisfying the absorption law
- Boolean algebra: a distributive, complemented lattice
Allowing axioms other than identities
One broadening of the concept of algebraic structure is to study sets with operations that must satisfy axioms other than identities.
- Integral domain: a ring with 0 ≠ 1 that has no zero divisors other than 0
- Division ring: an integral domain with an inverse operation (the inverse operation is not defined on the whole set)
- Field: a commutative division ring
Although these structures undoubtedly have an algebraic flavor, they suffer from defects not found in universal algebra. For example, there does not exist a product of two integral domains, nor a free field over any set.
Allowing additional structure
Algebraic structures can also be defined on sets with additional non-algebraic structures, such as topological spaces. The algebraic structure is required to be somehow compatible with the additional structure.
- Ordered group: a group with a compatible partial order
- Linearly ordered group: a group with a compatible linear order
- Archimedean group: a linearly ordered group for which the Archimedean property holds
- Topological group: a group with a compatible topology
- Lie group: a group with a compatible manifold structure
- Graded algebra: an algebra with a "grading"
- Clifford algebra: an associative algebra determined by quadratic form on a vector space
- Topological vector space: a vector space with a compatible topology
Categories
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.
There are various concepts in category theory that try to capture the algebraic character of a context, for instance
functors and categories.
See also
de:Algebraische Struktur fr:Structure algébrique it:Struttura algebrica he:מבנה אלגברי nl:Algebraïsche structuur ja:代数的構造 pt:Estrutura algébrica ru:Алгебраическая система sv:Algebraisk struktur uk:Алгебраїчна система zh:代数结构
