Subalgebra
Encyclopedia
In mathematics
, the word "algebra
", when referring to a structure, often means a vector space
or module
equipped with an additional bilinear operation. Algebras in universal algebra
are far more general: they are a common generalisation of all algebraic structures. In both contexts a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations.
is a vector subspace which is closed under the multiplication of vectors. The restriction of the algebra multiplication makes it an algebra over the same ring or field. This notion also applies to most specializations, where the multiplication must satisfy additional properties, e.g. to associative algebra
s or to Lie algebra
s. Only for unital algebras is there a stronger notion, of unital subalgebra, for which it is also required that the unit of the subalgebra be the unit of the bigger algebra.
, a subalgebra of an algebra
A is a subset
S of A that also has the structure of an algebra of the same type when the algebraic operations are restricted to S. If the axioms of a kind of algebraic structure
s is described by equational laws
, as is typically the case in universal algebra, then the only thing that needs to be checked is that S is closed
under the operations.
Some authors consider algebras with partial functions. There are various ways of defining subalgebras for these. Another generalization of algebras is to allow relations. These more general algebras are usually called structures
, and they are studied in model theory
and in theoretical computer science
. For structures with relations there are notions of weak and of induced substructure
s.
in universal algebra is (×, −1,1). (Inversion and unit are needed to get the right notions of homomorphism and so that the group laws can be expressed as equations.) Therefore a subgroup
of a group G is a subset S of G such that:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the word "algebra
Algebra (disambiguation)
Algebra, named for the Persian mathematician al Khwārizmī, describes one of the main branches of mathematics. It can also be used as a name for several different types of mathematical structures.-As a branch of mathematics:...
", when referring to a structure, often means a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
or module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
equipped with an additional bilinear operation. Algebras in universal algebra
Universal algebra
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
are far more general: they are a common generalisation of all algebraic structures. In both contexts a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations.
Subalgebras for algebras over a ring or field
A subalgebra of an algebra over a commutative ring or fieldAlgebra over a field
In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...
is a vector subspace which is closed under the multiplication of vectors. The restriction of the algebra multiplication makes it an algebra over the same ring or field. This notion also applies to most specializations, where the multiplication must satisfy additional properties, e.g. to associative algebra
Associative algebra
In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...
s or to Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
s. Only for unital algebras is there a stronger notion, of unital subalgebra, for which it is also required that the unit of the subalgebra be the unit of the bigger algebra.
Example
The 2×2-matrices over the reals form a unital algebra in the obvious way. The 2×2-matrices for which all entries are zero, except for the first one on the diagonal, form a subalgebra. It is also unital, but it is not a unital subalgebra.Subalgebras in universal algebra
In universal algebraUniversal algebra
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
, a subalgebra of an algebra
Structure (mathematical logic)
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations which are defined on it....
A is a subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
S of A that also has the structure of an algebra of the same type when the algebraic operations are restricted to S. If the axioms of a kind of algebraic structure
Algebraic structure
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...
s is described by equational laws
Variety (universal algebra)
In mathematics, specifically universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satisfying a given set of identities. Equivalently, a variety is a class of algebraic structures of the same signature which is closed under the taking of homomorphic...
, as is typically the case in universal algebra, then the only thing that needs to be checked is that S is closed
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
under the operations.
Some authors consider algebras with partial functions. There are various ways of defining subalgebras for these. Another generalization of algebras is to allow relations. These more general algebras are usually called structures
Structure (mathematical logic)
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations which are defined on it....
, and they are studied in model theory
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
and in theoretical computer science
Theoretical computer science
Theoretical computer science is a division or subset of general computer science and mathematics which focuses on more abstract or mathematical aspects of computing....
. For structures with relations there are notions of weak and of induced substructure
Substructure
In mathematical logic, an substructure or subalgebra is a structure whose domain is a subset of that of a bigger structure, and whose functions and relations are the traces of the functions and relations of the bigger structure...
s.
Example
For example, the standard signature for groupsGroup (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
in universal algebra is (×, −1,1). (Inversion and unit are needed to get the right notions of homomorphism and so that the group laws can be expressed as equations.) Therefore a subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
of a group G is a subset S of G such that:
- the identity e of G belongs to S (so that S is closed under the identity constant operation);
- whenever x belongs to S, so does x−1 (so that S is closed under the inverse operation);
- whenever x and y belong to S, so does x * y (so that S is closed under the group's multiplication operation).