Outline of algebraic structures

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In mathematics, many types of algebraic structures 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.

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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.

Study of algebraic structures

Algebraic structures appear in most branches of mathematics, and one can encounter them in many different ways.

Types of algebraic structures

In full generality, an algebraic structure may use any number of sets and any number of axioms in its definition. The most commonly studied structures, however, usually involve only one or two sets and one or two binary operations. The structures below are organized by how many sets are involved, and how many binary operations are used. Increased indentation is meant to indicate a more exotic structure, and the least indented levels are the most basic.

One set with no binary operations

One binary operation on one set

Group-like structures
Closure Associative Identity Cancellation Commutative
Partial magma UnneededUnneededUnneededUnneededUnneeded
Semigroupoid UnneededRequiredUnneededUnneededUnneeded
Small category UnneededRequiredRequiredUnneededUnneeded
Groupoid UnneededRequiredRequiredRequiredUnneeded
Commutative Groupoid UnneededRequiredRequiredRequiredRequired
Magma RequiredUnneededUnneededUnneededUnneeded
Commutative magma RequiredUnneededUnneededUnneededRequired
Quasigroup RequiredUnneededUnneededRequiredUnneeded
Commutative quasigroup RequiredUnneededUnneededRequiredRequired
Unital magma RequiredUnneededRequiredUnneededUnneeded
Commutative unital magma RequiredUnneededRequiredUnneededRequired
Loop RequiredUnneededRequiredRequiredUnneeded
Commutative loop RequiredUnneededRequiredRequiredRequired
Semigroup RequiredRequiredUnneededUnneededUnneeded
Commutative semigroup RequiredRequiredUnneededUnneededRequired
Associative quasigroup RequiredRequiredUnneededRequiredUnneeded
Commutative-and-associative quasigroup RequiredRequiredUnneededRequiredRequired
Monoid RequiredRequiredRequiredUnneededUnneeded
Commutative monoid RequiredRequiredRequiredUnneededRequired
Group RequiredRequiredRequiredRequiredUnneeded
Abelian group RequiredRequiredRequiredRequiredRequired

The following group-like structures consist of a set with a binary operation. The binary operation can be indicated by any symbol, or with no symbol (juxtaposition). The most common structure is that of a group. Other structures involve weakening or strengthening the axioms for groups, and may additionally use unary operations.

Two binary operations on one set

The main types of structures with one set having two binary operations are ring-like or ringoids and lattice-like or simply lattices. 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.

In ring-like structures or ringoids, the two binary operations are often called addition and multiplication, with multiplication linked to addition by the distributive law.

Lattice-like structures have two binary operations called meet and join, connected by the absorption law.

Module-like structures on two sets

The following module-like structures have the common feature of having two sets, A and B, so that there is a binary operation from A×A into A and another operation from A×B into A. Modules, counting the ring operations, have at least three binary operations.

Algebra-like structures on two sets

These structures are 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. Many of these structures are hybrid structures of the previously mentioned ones.

Algebraic structures with additional non-algebraic structure

There are many examples of mathematical structures where algebraic structure exists alongside non-algebraic structure.

Algebraic structures in different disciplines

Some algebraic structures find uses in disciplines outside of abstract algebra. The following is meant to demonstrate some specific applications in other fields.

In Physics:

In Mathematical logic:

In Computer science:

See also

Related Research Articles

In mathematics, an associative algebraA over a commutative ring K is a ring A together with a ring homomorphism from K into the center of A. This is thus an algebraic structure with an addition, a multiplication, and a scalar multiplication. The addition and multiplication operations together give A the structure of a ring; the addition and scalar multiplication operations together give A the structure of a module or vector space over K. In this article we will also use the term K-algebra to mean an associative algebra over K. A standard first example of a K-algebra is a ring of square matrices over a commutative ring K, with the usual matrix multiplication.

In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type. The word homomorphism comes from the Ancient Greek language: ὁμός meaning "same" and μορφή 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". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925).

<span class="mw-page-title-main">Monoid</span> Algebraic structure with an associative operation and an identity element

In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.

<span class="mw-page-title-main">Semigroup</span> Algebraic structure consisting of a set with an associative binary operation

In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.

Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, in universal algebra one takes the class of groups as an object of study.

In abstract algebra, a congruence relation is an equivalence relation on an algebraic structure 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 for the relation.

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. Informally, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

In mathematics, an algebraic structure consists of a nonempty set A, a collection of operations on A, and a finite set of identities that these operations must satisfy.

In abstract algebra, a magma, binar, or, rarely, groupoid is a basic kind of algebraic structure. Specifically, a magma consists of a set equipped with a single binary operation that must be closed by definition. No other properties are imposed.

In mathematics, an algebra over a field is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear".

Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject.

In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of module also generalizes the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers.

In abstract algebra, a semiring is an algebraic structure. Semirings are a generalization of rings, dropping the requirement that each element must have an additive inverse. At the same time, semirings are a generalization of bounded distributive lattices.

In abstract algebra, a medial magma or medial groupoid is a magma or groupoid (that is, a set with a binary operation) that satisfies the identity

In abstract algebra, a residuated lattice is an algebraic structure that is simultaneously a lattice xy and a monoid xy which admits operations x\z and z/y, loosely analogous to division or implication, when xy is viewed as multiplication or conjunction, respectively. Called respectively right and left residuals, these operations coincide when the monoid is commutative. The general concept was introduced by Morgan Ward and Robert P. Dilworth in 1939. Examples, some of which existed prior to the general concept, include Boolean algebras, Heyting algebras, residuated Boolean algebras, relation algebras, and MV-algebras. Residuated semilattices omit the meet operation ∧, for example Kleene algebras and action algebras.

A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure A is a non-associative algebra over a field K if it is a vector space over K and is equipped with a K-bilinear binary multiplication operation A × AA which may or may not be associative. Examples include Lie algebras, Jordan algebras, the octonions, and three-dimensional Euclidean space equipped with the cross product operation. Since it is not assumed that the multiplication is associative, using parentheses to indicate the order of multiplications is necessary. For example, the expressions (ab)(cd), (a(bc))d and a(b(cd)) may all yield different answers.

In mathematics, a semifield is an algebraic structure with two binary operations, addition and multiplication, which is similar to a field, but with some axioms relaxed.

<span class="mw-page-title-main">Abstract algebra</span> Branch of mathematics

In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations acting on their elements. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term abstract algebra was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except in pedagogy.

References

  1. Jonathan D. H. Smith (15 November 2006). An Introduction to Quasigroups and Their Representations. Chapman & Hall. ISBN   9781420010633 . Retrieved 2012-08-02.

A monograph available free online: