stayAbstract algebraAlgebraic structure refers to one or moreoperation(In general, infinitely many operations are allowed[1]）OfNon empty set。The algebraic structures generally studied aregroup、ring、field、grid、model, field algebra andvector space wait.In mathematics, more specifically, in abstract algebra, algebraic structure is a set (called carrier set or underlying set), which defines one or more finite operations that satisfy axioms.
Algebraic structure refers to many mathematical objects, such as groupsring、field、vector space 、Ordered setFirst, because of the diversity of mathematical objects, there are different types of sets. For example, the set represented by a group is G × G. In fact, a group involves binary operations;The set represented by vector space is F × F → F, F × V → V, V × V → V. Vector space involves operations in field F, operations of elements in field F on elements in V, and operations of elements in V. Introduce the basic concept of "composition" (for example, the composition of groups is multiplication operations; the "composition" of vector space involves multiplication of elements in F on elements in V,Addition of elements in V), and "composition" is required to be suitable for the given axiom system, which results in a mathematical structure.
For example, group 〈 G, * 〉 is a mathematical structure. The mapping * (composition or algebraic operation) from set G, G × G to G is suitable for
(a*b)*c=a*(b*c) a，b，c∈G
∃1∈G，1*a=a*1=a，
a∈G
a∈G，∃a′∈G，a′*a=a* a′=1.
In fact, in algebraic structure, all concepts can be defined by sets and relations, that is, expressed in the language of sets and relations.
As a basic concept, if we only focus on "composition" (that is, "operation"), this mathematical structure is called algebraic structure, or algebraic system (system). In other words, algebraic structure (algebraic system) is a set with several compositions (operations).
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Examples of algebraic structures include groups, rings, fields, andgrid。More complex structures can be defined by introducing multiple operations, different underlying collections, or modifying definition axioms.Examples of more complex algebraic structures include vector spaces, modules, and algebra.A group is an algebraic structure with a binary operation;Ring and field are algebraic structures with two binary operations;Lattice is an algebraic structure with two binary operations;Boolean algebra, set algebra and propositional algebra are all algebraic structures with two binary operations and one unary operation. They are (respectively) suitable for specific axiom systems.
The properties of specific algebraic structure are studied in abstract algebra.The general theory of algebraic structure has been formalized in general algebra.The language of category theory is used to express and study the relationship between different types of algebraic and non algebraic objects.This is because sometimes strong connections may be found between objects of certain classes, sometimes among objects of different classes.For example, Galois theory established the connection between some fields and groups: two different types of algebraic structures.
Addition and multiplication on numbers is a typical example of an operation, which combines two elements of a set to produce a third element.These operations obey several algebraic laws.For example, a+(b+c)=(a+b)+c and a (bc)=(ab) c are both incidence laws.And a+b=b+a, ab=ba, commutative law.Many systems studied by mathematicians have operations that follow some, but not necessarily all, common arithmetic rules.For example, you can combine the rotation of objects in 3D space by performing the first rotation, and then apply the second rotation to objects in their new direction.This rotation operation obeys the correlation law, but can invalidate the exchange law.
Mathematicians give the name of a set that has one or more operations subject to a specific set of laws, and study them abstractly as algebraic structures.When a new problem can be proved to follow the law of one of these algebraic structures, all the work done in the past on this kind of problem can be applied to the new problem.
In a completely universal case, algebraic structures may involve any number of sets and operations, which can combine more than two elements (higher arity), but this article focuses on one or two sets of binary operations.The examples here are by no means a complete list, but they are representative and contain the most common structures.A longer list of algebraic structures can be found in external links and classes: algebraic structures.Structures are listed in approximate order of complexity.
Hybrid structure
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Algebraic structures can also coexist with structures with additional non algebraic properties, such as partial order or topology.In a sense, additional structures must be compatible with algebraic structures.[2]
Topology group: Topology group consistent with the group operation.
Lie group: a topological group with consistent smooth manifold structure.
Ordered groups, ordered rings and ordered fields: each class of structures that are locally ordered.
Archimedean group: Linear ordered group with Archimedean properties.
Normed vector space : A vector space with consistent norms.If such a space is complete (as a metric space), it is called a Banach space.
Hilbert spaceThe inner product of an inner product space on a real or complex number produces a Banach space structure.
Von Neumann Algebra: An Algebra with Weak Operator TopologyHilbert spaceAlgebra of upper operator.
General algebra
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Algebraic structure is defined by different axiomatic configurations.General algebra studies these objects abstractly.A major dichotomy is divided into: structures that are completely defined by themselves and structures that cannot be completely defined by themselves.If all axioms defining a class of algebra are identities, then the class of the object has a variety (algebraic diversity in algebraic geometry).[3]
There is a symbol containing two operators in the group: multiplication operator m with two parameters, inverse operator i with one parameter, and identification element constant e (it can be considered as an operator with zero parameters).Given a set of variables (numbers) such as x, y, z, etc., algebra is the set of all possible m, i, e and variables, for example, m (i (X), m (x, m (y, e) is algebra, and one of the axioms defining a group is m (x, i (X)=e;Another axiom is that m (x, e)=x. The axiom can be expressed as a tree.These equations derive equivalence classes from free algebras, and quotient algebras have the algebraic structure of groups.
Categorical theory
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Category theory is another tool for studying algebraic structure (for example, macLane in 2003).A category is a collection of objects with associated morphisms.Each algebraic structure has its own concept of homomorphism, that is, any function compatible with the operations that define the structure.Therefore, each algebraic structure will generate a category.For example, the category of groups is to take all groups as objects, and all group homomorphisms as morphisms.This specific category can be regarded as a set of categories with the theoretical structure of additional categories.Similarly, the category of topological groups (whose morphisms are homomorphisms of continuous groups) is a category of topological spaces with additional structures.[4]
The relevant concepts of category theory are as follows:
