Algebraic structure

In mathematics, and more specifically in abstract algebra, an algebraic structure on a set A (called carrier set or underlying set) is a collection of finitary operations on A; the set A with this structure is also called an algebra.[1]

Examples of algebraic structures include groups, rings, fields, and lattices. More complex structures can be defined by introducing multiple operations, different underlying sets, or by altering the defining axioms. Examples of more complex algebraic structures include vector spaces, modules, and algebras.

The properties of specific algebraic structures are studied in abstract algebra. The general theory of algebraic structures has been formalized in universal algebra. The language of category theory is used to express and study relationships between different classes of algebraic and non-algebraic objects. This is because it is sometimes possible to find strong connections between some classes of objects, sometimes of different kinds. For example, Galois theory establishes a connection between certain fields and groups: two algebraic structures of different kinds.

Introduction

Addition and multiplication on numbers are the prototypical example of an operation that combines two elements of a set to produce a third. These operations obey several algebraic laws. For example, a + (b + c) = (a + b) + c and a(bc) = (ab)c, both examples of the associative law. Also a + b = b + a, and ab = ba, the commutative law. Many systems studied by mathematicians have operations that obey some, but not necessarily all, of the laws of ordinary arithmetic. For example, rotations of objects in three-dimensional space can be combined by performing the first rotation and then applying the second rotation to the object in its new orientation. This operation on rotations obeys the associative law, but can fail the commutative law.

Mathematicians give names to sets with one or more operations that obey a particular collection of laws, and study them in the abstract as algebraic structures. When a new problem can be shown to follow the laws of one of these algebraic structures, all the work that has been done on that category in the past can be applied to the new problem.

In full generality, algebraic structures may involve an arbitrary number of sets and operations that can combine more than two elements (higher arity), but this article focuses on binary operations on one or two sets. The examples here are by no means a complete list, but they are meant to be a representative list and include the most common structures. Longer lists of algebraic structures may be found in the external links and within Category:Algebraic structures. Structures are listed in approximate order of increasing complexity.

Examples

One set with operations

Simple structures: no binary operation:

  • Set: a degenerate algebraic structure S having no operations.
  • Pointed set: S has one or more distinguished elements, often 0, 1, or both.
  • Unary system: S and a single unary operation over S.
  • Pointed unary system: a unary system with S a pointed set.

Group-like structures: one binary operation. The binary operation can be indicated by any symbol, or with no symbol (juxtaposition) as is done for ordinary multiplication of real numbers.

Ring-like structures or Ringoids: two binary operations, often called addition and multiplication, with multiplication distributing over addition.

  • Semiring: a ringoid such that S is a monoid under each operation. Addition is typically assumed to be commutative and associative, and the monoid product is assumed to distribute over the addition on both sides, and the additive identity satisfies 0 x = 0 for all x.
  • Near-ring: a semiring whose additive monoid is a (not necessarily abelian) group.
  • Ring: a semiring whose additive monoid is an abelian group.
  • Lie ring: a ringoid whose additive monoid is an abelian group, but whose multiplicative operation satisfies the Jacobi identity rather than associativity.
  • Boolean ring: a commutative ring with idempotent multiplication operation.
  • Field: a commutative ring which contains a multiplicative inverse for every nonzero element
  • Kleene algebras: a semiring with idempotent addition and a unary operation, the Kleene star, satisfying additional properties.
  • *-algebra: a ring with an additional unary operation (*) satisfying additional properties.

Lattice structures: two or more binary operations, including operations called meet and join, connected by the absorption law.[3]

  • Complete lattice: a lattice in which arbitrary meet and joins exist.
  • Bounded lattice: a lattice with a greatest element and least element.
  • Complemented lattice: a bounded lattice with a unary operation, complementation, denoted by postfix . The join of an element with its complement is the greatest element, and the meet of the two elements is the least element.
  • Modular lattice: a lattice whose elements satisfy the additional modular identity.
  • Distributive lattice: a lattice in which each of meet and join distributes over the other. Distributive lattices are modular, but the converse does not hold.
  • Boolean algebra: a complemented distributive lattice. Either of meet or join can be defined in terms of the other and complementation. This can be shown to be equivalent with the ring-like structure of the same name above.
  • Heyting algebra: a bounded distributive lattice with an added binary operation, relative pseudo-complement, denoted by infix →, and governed by the axioms x → x = 1, x (x → y) = x y, y (x → y) = y, x → (y z) = (x → y) (x → z).

Arithmetics: two binary operations, addition and multiplication. S is an infinite set. Arithmetics are pointed unary systems, whose unary operation is injective successor, and with distinguished element 0.

  • Robinson arithmetic. Addition and multiplication are recursively defined by means of successor. 0 is the identity element for addition, and annihilates multiplication. Robinson arithmetic is listed here even though it is a variety, because of its closeness to Peano arithmetic.
  • Peano arithmetic. Robinson arithmetic with an axiom schema of induction. Most ring and field axioms bearing on the properties of addition and multiplication are theorems of Peano arithmetic or of proper extensions thereof.

Two sets with operations

Module-like structures: composite systems involving two sets and employing at least two binary operations.

  • Group with operators: a group G with a set Ω and a binary operation Ω × GG satisfying certain axioms.
  • Module: an abelian group M and a ring R acting as operators on M. The members of R are sometimes called scalars, and the binary operation of scalar multiplication is a function R × MM, which satisfies several axioms. Counting the ring operations these systems have at least three operations.
  • Vector space: a module where the ring R is a division ring or field.
  • Graded vector space: a vector space with a direct sum decomposition breaking the space into "grades".
  • Quadratic space: a vector space V over a field F with a function from V into F satisfying certain properties. Every quadratic space is also an inner product space (see below).

Algebra-like structures: composite system defined over two sets, a ring R and an R-module M equipped with an operation called multiplication. This can be viewed as a system with five binary operations: two operations on R, two on M and one involving both R and M.

  • Algebra over a ring (also R-algebra): a module over a commutative ring R, which also carries a multiplication operation that is compatible with the module structure. This includes distributivity over addition and linearity with respect to multiplication by elements of R. The theory of an algebra over a field is especially well developed.
  • Associative algebra: an algebra over a ring such that the multiplication is associative.
  • Nonassociative algebra: a module over a commutative ring, equipped with a ring multiplication operation that is not necessarily associative. Often associativity is replaced with a different identity, such as alternation, the Jacobi identity, or the Jordan identity.
  • Coalgebra: a vector space with a "comultiplication" defined dually to that of associative algebras.
  • Lie algebra: a special type of nonassociative algebra whose product satisfies the Jacobi identity.
  • Lie coalgebra: a vector space with a "comultiplication" defined dually to that of Lie algebras.
  • Graded algebra: a graded vector space with an algebra structure compatible with the grading. The idea is that if the grades of two elements a and b are known, then the grade of ab is known, and so the location of the product ab is determined in the decomposition.
  • Inner product space: an F vector space V with a sesquilinear binary operation V × VF.

Four or more binary operations:

Hybrid structures

Algebraic structures can also coexist with added structure of non-algebraic nature, such as partial order or a topology. The added structure must be compatible, in some sense, with the algebraic structure.

Universal algebra

Algebraic structures are defined through different configurations of axioms. Universal algebra abstractly studies such objects. One major dichotomy is between structures that are axiomatized entirely by identities and structures that are not. If all axioms defining a class of algebras are identities, then this class is a variety (not to be confused with algebraic varieties of algebraic geometry).

Identities are equations formulated using only the operations the structure allows, and variables that are tacitly universally quantified over the relevant universe. Identities contain no connectives, existentially quantified variables, or relations of any kind other than the allowed operations. The study of varieties is an important part of universal algebra. An algebraic structure in a variety may be understood as the quotient algebra of term algebra (also called "absolutely free algebra") divided by the equivalence relations generated by a set of identities. So, a collection of functions with given signatures generate a free algebra, the term algebra T. Given a set of equational identities (the axioms), one may consider their symmetric, transitive closure E. The quotient algebra T/E is then the algebraic structure or variety. Thus, for example, groups have a signature containing two operators: the multiplication operator m, taking two arguments, and the inverse operator i, taking one argument, and the identity element e, a constant, which may be considered an operator that takes zero arguments. Given a (countable) set of variables x, y, z, etc. the term algebra is the collection of all possible terms involving m, i, e and the variables; so for example, m(i(x), m(x,m(y,e))) would be an element of the term algebra. One of the axioms defining a group is the identity m(x, i(x)) = e; another is m(x,e) = x. The axioms can be represented as trees. These equations induce equivalence classes on the free algebra; the quotient algebra then has the algebraic structure of a group.

Some structures do not form varieties, because either:

  1. It is necessary that 0 ≠ 1, 0 being the additive identity element and 1 being a multiplicative identity element, but this is a nonidentity;
  2. Structures such as fields have some axioms that hold only for nonzero members of S. For an algebraic structure to be a variety, its operations must be defined for all members of S; there can be no partial operations.

Structures whose axioms unavoidably include nonidentities are among the most important ones in mathematics, e.g., fields and division rings. Structures with nonidentities present challenges varieties do not. For example, the direct product of two fields is not a field, because , but fields do not have zero divisors.

Category theory

Category theory is another tool for studying algebraic structures (see, for example, Mac Lane 1998). A category is a collection of objects with associated morphisms. Every algebraic structure has its own notion of homomorphism, namely any function compatible with the operation(s) defining the structure. In this way, every algebraic structure gives rise to a category. For example, the category of groups has all groups as objects and all group homomorphisms as morphisms. This concrete category may be seen as a category of sets with added category-theoretic structure. Likewise, the category of topological groups (whose morphisms are the continuous group homomorphisms) is a category of topological spaces with extra structure. A forgetful functor between categories of algebraic structures "forgets" a part of a structure.

There are various concepts in category theory that try to capture the algebraic character of a context, for instance

Different meanings of "structure"

In a slight abuse of notation, the word "structure" can also refer to just the operations on a structure, instead of the underlying set itself. For example, the sentence, "We have defined a ring structure on the set ," means that we have defined ring operations on the set . For another example, the group can be seen as a set that is equipped with an algebraic structure, namely the operation .

See also

Notes

  1. ^ P.M. Cohn. (1981) Universal Algebra, Springer, p. 41.
  2. ^ Jonathan D. H. Smith. An Introduction to Quasigroups and Their Representations. Chapman & Hall. Retrieved 2012-08-02.
  3. ^ Ringoids and lattices can be clearly distinguished despite both having two defining binary operations. In the case of ringoids, the two operations are linked by the distributive law; in the case of lattices, they are linked by the absorption law. Ringoids also tend to have numerical models, while lattices tend to have set-theoretic models.

References

Category theory

External links

Boolean domain

In mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include false and true. In logic, mathematics and theoretical computer science, a Boolean domain is usually written as {0, 1}, {false, true}, {F, T}, or

The algebraic structure that naturally builds on a Boolean domain is the Boolean algebra with two elements. The initial object in the category of bounded lattices is a Boolean domain.

In computer science, a Boolean variable is a variable that takes values in some Boolean domain. Some programming languages feature reserved words or symbols for the elements of the Boolean domain, for example false and true. However, many programming languages do not have a Boolean datatype in the strict sense. In C or BASIC, for example, falsity is represented by the number 0 and truth is represented by the number 1 or −1, and all variables that can take these values can also take any other numerical values.

Centrosymmetric matrix

In mathematics, especially in linear algebra and matrix theory, a centrosymmetric matrix is a matrix which is symmetric about its center. More precisely, an n × n matrix A = [ Ai,j ] is centrosymmetric when its entries satisfy

Ai,j = An−i+1,n−j+1 for 1 ≤ i,j ≤ n.If J denotes the n × n matrix with 1 on the counterdiagonal and 0 elsewhere (that is, Ji,n+1-i = 1; Ji,j = 0 if j ≠ n+1-i), then a matrix A is centrosymmetric if and only if AJ = JA. The matrix J is sometimes referred to as the exchange matrix.

Chiral algebra

In mathematics, a chiral algebra is an algebraic structure introduced by Beilinson & Drinfeld (2004) as a rigorous version of the rather vague concept of a chiral algebra in physics.

Congruence relation

In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done with equivalent elements will yield equivalent elements. Every congruence relation has a corresponding quotient structure, whose elements are the equivalence classes (or congruence classes) for the relation.

Direct product of groups

In group theory, the direct product is an operation that takes two groups G and H and constructs a new group, usually denoted G × H. This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics.

In the context of abelian groups, the direct product is sometimes referred to as the direct sum, and is denoted G ⊕ H. Direct sums play an important role in the classification of abelian groups: according to the fundamental theorem of finite abelian groups, every finite abelian group can be expressed as the direct sum of cyclic groups.

Equivalence relation

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The relation "is equal to" is the canonical example of an equivalence relation, where for any objects a, b, and c:

a = a (reflexive property),

if a = b then b = a (symmetric property), and

if a = b and b = c then a = c (transitive property).As a consequence of the reflexive, symmetric, and transitive properties, any equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class.

Free object

In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. It is a part of universal algebra, in the sense that it relates to all types of algebraic structure (with finitary operations). It also has a formulation in terms of category theory, although this is in yet more abstract terms. Examples include free groups, tensor algebras, or free lattices. Informally, a free object over a set A can be thought of as being a "generic" algebraic structure over A: the only equations that hold between elements of the free object are those that follow from the defining axioms of the algebraic structure.

Half-integer

In mathematics, a half-integer is a number of the form

,

where is an integer. For example,

4½, 7/2, −13/2, 8.5

are all half-integers.

Half-integers occur frequently enough in mathematical contexts that a special term for them is convenient. Note that a half of an integer is not always a half-integer: half of an even integer is an integer but not a half-integer. The half-integers are precisely those numbers that are half of an odd integer, and for this reason are also called the half-odd-integers. Half-integers are a special case of the dyadic rationals, numbers that can be formed by dividing an integer by a power of two.

Homomorphism

In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word homomorphism comes from the ancient Greek language: ὁμός (homos) meaning "same" and μορφή (morphe) meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German ähnlich meaning "similar" to ὁμός meaning "same".Homomorphisms of vector spaces are also called linear maps, and their study is the object of linear algebra.

The concept of homomorphism has been generalized, under the name of morphism, to many other structures that either do not have an underlying set, or are not algebraic. This generalization is the starting point of category theory.

A homomorphism may also be an isomorphism, an endomorphism, an automorphism, etc. (see below). Each of those can be defined in a way that may be generalized to any class of morphisms.

Jacobian variety

In mathematics, the Jacobian variety J(C) of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of C, hence an abelian variety.

List of finite spherical symmetry groups

Finite spherical symmetry groups are also called point groups in three dimensions. There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral, octahedral, and icosahedral symmetry.

This article lists the groups by Schoenflies notation, Coxeter notation, orbifold notation, and order. John Conway uses a variation of the Schoenflies notation, based on the groups' quaternion algebraic structure, labeled by one or two upper case letters, and whole number subscripts. The group order is defined as the subscript, unless the order is doubled for symbols with a plus or minus, "±", prefix, which implies a central inversion.Hermann–Mauguin notation (International notation) is also given. The crystallography groups, 32 in total, are a subset with element orders 2, 3, 4 and 6.

Magma (computer algebra system)

Magma is a computer algebra system designed to solve problems in algebra, number theory, geometry and combinatorics. It is named after the algebraic structure magma. It runs on Unix-like operating systems, as well as Windows.

Near field

Near field may refer to:

Near-field (mathematics), an algebraic structure

Near-field region, part of an electromagnetic field

Near field (Electromagnetism)

Magnetoquasistatic field, the magnetic component of the electromagnetic near field

Near-field communication (NFC) using the magnetic component of the electromagnetic near field (Magnetoquasistatic field)

Outline of algebraic structures

In mathematics, there are many types of algebraic structures which are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures may be viewed in different ways, however the common starting point of algebra texts is that an algebraic object incorporates one or more sets with one or more binary operations or unary operations satisfying a collection of axioms.

Another branch of mathematics known as universal algebra studies algebraic structures in general. From the universal algebra viewpoint, most structures can be divided into varieties and quasivarieties depending on the axioms used. Some axiomatic formal systems that are neither varieties nor quasivarieties, called nonvarieties, are sometimes included among the algebraic structures by tradition.

Concrete examples of each structure will be found in the articles listed.

Algebraic structures are so numerous today that this article will inevitably be incomplete. In addition to this, there are sometimes multiple names for the same structure, and sometimes one name will be defined by disagreeing axioms by different authors. Most structures appearing on this page will be common ones which most authors agree on. Other web lists of algebraic structures, organized more or less alphabetically, include Jipsen and PlanetMath. These lists mention many structures not included below, and may present more information about some structures than is presented here.

Perfect square

Perfect square is an element of algebraic structure which is equal to the square of another element.

Square number, a perfect square integer

Quotient (universal algebra)

In mathematics, a quotient algebra is the result of partitioning the elements of an algebraic structure using a congruence relation.

Quotient algebras are also called factor algebras. Here, the congruence relation must be an equivalence relation that is additionally compatible with all the operations of the algebra, in the formal sense described below.

Its equivalence classes partition the elements of the given algebraic structure. The quotient algebra has these classes as its elements, and the compatibility conditions are used to give the classes an algebraic structure.The idea of the quotient algebra abstracts into one common notion the quotient structure of quotient rings of ring theory, quotient groups of group theory, the quotient spaces of linear algebra and the quotient modules of representation theory into a common framework.

Reduct

In universal algebra and in model theory, a reduct of an algebraic structure is obtained by omitting some of the operations and relations of that structure. The converse of "reduct" is "expansion."

Signature (logic)

In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are used for both purposes.

Signatures play the same role in mathematics as type signatures in computer programming. They are rarely made explicit in more philosophical treatments of logic.

Substitution (algebra)

In algebra, the operation of substitution can be applied in various contexts involving formal objects containing symbols (often called variables or indeterminates); the operation consists of systematically replacing occurrences of some symbol by a given value.

Substitution is a basic operation of computer algebra. It is generally called "subs" or "subst" in computer algebra systems.

A common case of substitution involves polynomials, where substitution of a numerical value for the indeterminate of a (univariate) polynomial amounts to evaluating the polynomial at that value. Indeed this operation occurs so frequently that the notation for polynomials is often adapted to it; instead of designating a polynomial by a name like P, as one would do for other mathematical objects, one could define

so that substitution for X can be designated by replacement inside "P(X)", say

or

.

Substitution can however also be applied to other kinds of formal objects built from symbols, for instance elements of free groups. In order for substitution to be defined, one needs an algebraic structure with an appropriate universal property, that asserts the existence of unique homomorphisms that send indeterminates to specific values; the substitution then amounts to finding the image under such a homomorphism.

Substitution is related to, but not identical to, function composition; it is also closely related to β-reduction in lambda calculus. In contrast to these notions, however, the accent in algebra is on the preservation of algebraic structure by the substitution operation, the fact that substitution gives a homomorphism for the structure at hand (in the case of polynomials, the ring structure).

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