In mathematics, there are many types of

algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...

s which are studied.

Abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...

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 operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary op ...

s or

unary operation In mathematics, an unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function , where is a set. The function is a unary operation ...

s satisfying a collection of

axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy o ...

s. Another branch of mathematics known as

universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular Group (mathematics), groups as ...

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

atic

formal system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A for ...

s 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, includ
Jipsen
an
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. *Beginning study: In American universities, groups,

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...

s and fields are generally the first structures encountered in subjects such as

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matric ...

. They are usually introduced as sets with certain axioms. *Advanced study: **

studies properties of specific algebraic structures. **

Universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular Group (mathematics), groups as ...

studies algebraic structures abstractly, rather than specific types of structures. *** Varieties ** Category theory studies interrelationships between different structures, algebraic and non-algebraic. To study a non-algebraic object, it is often useful to use category theory to relate the object to an algebraic structure. ***Example: The

fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...

of a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...

gives information about the topological space.

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

s. 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

* Set: a degenerate algebraic structure ''S'' having no operations. *

Pointed set In mathematics, a pointed set (also based set or rooted set) is an ordered pair (X, x_0) where X is a set and x_0 is an element of X called the base point, also spelled basepoint. Maps between pointed sets (X, x_0) and (Y, y_0) – called based ma ...

: ''S'' has one or more distinguished elements, often 0, 1, or both. * Unary system: ''S'' and a single

over ''S''. * : a unary system with ''S'' a pointed set.

One binary operation on one set

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. * Magma or groupoid: ''S'' and a single binary operation over ''S''. *

Semigroup In mathematics, a semigroup is an algebraic structure consisting of a Set (mathematics), set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplication, multiplicatively ...

: an

associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...

magma. *

Monoid 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. Monoids ...

: a semigroup with

identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures s ...

. * Group: a monoid with a unary operation (inverse), giving rise to

inverse element In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...

s. **

Abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is com ...

: a group whose binary operation is

commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...

. *

Quasigroup In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that " division" is always possible. Quasigroups differ from groups mainly in that they need not be associative and need not h ...

: a magma obeying the Latin square property. A quasigroup may also be represented using three binary operations. *

Loop Loop or LOOP may refer to: Brands and enterprises * Loop (mobile), a Bulgarian virtual network operator and co-founder of Loop Live * Loop, clothing, a company founded by Carlos Vasquez in the 1990s and worn by Digable Planets * Loop Mobile, ...

: a quasigroup with identity. *

Semilattice In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilattice) is a partially ordered set which has ...

: a semigroup whose operation is

idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...

and commutative. The binary operation can be called either meet or join. This is basically "half" of a lattice structure (see below).

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 mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmetic ...

; 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 Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...

models. In ring-like structures or ringoids, the two binary operations are often called

addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or ''sum'' of ...

and

multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being ad ...

, with multiplication linked to addition by the

. *

Semiring In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. The term rig is also used occasionally—this originated as a joke, suggesting that rigs a ...

: 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 0 is an

absorbing element In mathematics, an absorbing element (or annihilating element) is a special type of element of a set with respect to a binary operation on that set. The result of combining an absorbing element with any element of the set is the absorbing element ...

in the sense that 0 ''x'' = 0 for all ''x''. *

Near-ring In mathematics, a near-ring (also near ring or nearring) is an algebraic structure similar to a ring but satisfying fewer axioms. Near-rings arise naturally from functions on groups. Definition A set ''N'' together with two binary operatio ...

: a semiring whose additive monoid is a (not necessarily abelian) group. * Ring: a semiring whose additive monoid is an abelian group. ** Commutative ring: a ring in which the multiplication operation is commutative. **

Division ring In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...

: a nontrivial ring in which division by nonzero elements is defined. **

Integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...

: A nontrivial commutative ring in which the product of any two nonzero elements is nonzero. ** Field: a commutative division ring (i.e. a commutative ring which contains a multiplicative inverse for every nonzero element). * Nonassociative rings: These are like rings, but the multiplication operation need not be associative. ** Lie ring: a ringoid whose additive monoid is an abelian group, but whose multiplicative operation satisfies the

Jacobi identity In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the assoc ...

rather than associativity. **

Jordan ring In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms: # xy = yx (commutative law) # (xy)(xx) = x(y(xx)) (). The product of two elements ''x'' and ''y'' in a Jordan alg ...

: a commutative nonassociative ring that respects the

Jordan identity In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms: # xy = yx ( commutative law) # (xy)(xx) = x(y(xx)) (). The product of two elements ''x'' and ''y'' in a Jordan ...

Boolean ring In mathematics, a Boolean ring ''R'' is a ring for which ''x''2 = ''x'' for all ''x'' in ''R'', that is, a ring that consists only of idempotent elements. An example is the ring of integers modulo 2. Every Boolean ring gives rise to a Boolean a ...

: a commutative ring with idempotent multiplication operation. * Kleene algebras: a semiring with idempotent addition and a unary operation, the

Kleene star In mathematical logic and computer science, the Kleene star (or Kleene operator or Kleene closure) is a unary operation, either on sets of strings or on sets of symbols or characters. In mathematics, it is more commonly known as the free monoid ...

, satisfying additional properties. * *-algebra or *-ring: a ring with an additional unary operation (*) known as an involution, satisfying additional properties. * Arithmetic: addition and multiplication on an

infinite set In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. Properties The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only ...

, with an additional pointed unary structure. The

injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contraposi ...

successor, and has distinguished element 0. **

Robinson arithmetic In mathematics, Robinson arithmetic is a finitely axiomatized fragment of first-order Peano arithmetic (PA), first set out by R. M. Robinson in 1950. It is usually denoted Q. Q is almost PA without the axiom schema of mathematical induction. Q is ...

. 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 In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly ...

. Robinson arithmetic with an

axiom schema In mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom. Formal definition An axiom schema is a formula in the metalanguage of an axiomatic system, in which one or more schematic variables ...

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. Lattice-like structures have two binary operations called meet and join, connected by the absorption law. * Latticoid: meet and join commute but need not associate. *

Skew lattice In abstract algebra, a skew lattice is an algebraic structure that is a non-commutative generalization of a lattice. While the term ''skew lattice'' can be used to refer to any non-commutative generalization of a lattice, since 1989 it has been use ...

: meet and join associate but need not commute. * Lattice: meet and join associate and commute. **

Complete lattice In mathematics, a complete lattice is a partially ordered set in which ''all'' subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a ''conditionally complete lattice.'' S ...

: a lattice in which arbitrary meet and joins exist. **

Bounded lattice A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bou ...

: a lattice with a

greatest element In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an el ...

and least element. **

Complemented lattice In the mathematical discipline of order theory, a complemented lattice is a bounded lattice (with least element 0 and greatest element 1), in which every element ''a'' has a complement, i.e. an element ''b'' satisfying ''a'' ∨ ''b''&n ...

: a bounded lattice with a unary operation, complementation, denoted by postfix ^{⊥
The up tack or falsum (⊥, \bot in LaTeX, U+22A5 in Unicode) is a constant symbol used to represent:

* The truth value 'false', or a logical constant denoting a proposition in logic that is always false (often called "falsum" or "absurdum").
* T ...}. 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 In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set ...

: 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 In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas ...

: 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 In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation ''a'' → ''b'' of '' ...

: a bounded distributive lattice with an added binary operation,

relative pseudo-complement In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation ''a'' → ''b'' of '' ...

, denoted by

infix An infix is an affix inserted inside a word stem (an existing word or the core of a family of words). It contrasts with '' adfix,'' a rare term for an affix attached to the outside of a stem, such as a prefix or suffix. When marking text for i ...

→, and governed by the axioms: *** ''x'' → ''x'' = 1 *** ''x'' (''x'' → ''y'') = ''x y'' *** ''y'' (''x'' → ''y'') = ''y'' *** ''x'' → (''y z'') = (''x'' → ''y'') (''x'' → ''z'')

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. * Group with operators: a group ''G'' with a set Ω and a binary operation Ω × ''G'' → ''G'' 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'' × ''M'' → ''M'', which satisfies several axioms. **Special types of modules, including

free module In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a fie ...

projective module In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characterizati ...

injective module In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module ''Q'' that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if ''Q'' is a submodule ...

s and

flat module In algebra, a flat module over a ring ''R'' is an ''R''- module ''M'' such that taking the tensor product over ''R'' with ''M'' preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact s ...

s are studied in abstract algebra. *

Vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...

s: A module where the ring ''R'' is a

division ring In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...

or field. **

Graded vector space In mathematics, a graded vector space is a vector space that has the extra structure of a '' grading'' or a ''gradation'', which is a decomposition of the vector space into a direct sum of vector subspaces. Integer gradation Let \mathbb be ...

s: Vector spaces which are equipped with a

direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mo ...

decomposition into subspaces or "grades". ** Quadratic space: a vector space ''V'' over a field ''F'' with a

quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...

on ''V'' taking values in ''F''.

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. *

Algebra over a ring In mathematics, an algebra over a field (often simply called an algebra) 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 additio ...

(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 Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...

with respect to multiplication by elements of ''R''. **

Algebra over a field In mathematics, an algebra over a field (often simply called an algebra) 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 additio ...

: This is a ring which is also a vector space over a field. Multiplication is usually assumed to be associative. The theory is especially well developed. * Associative algebra: an algebra over a ring such that the multiplication is

. * 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

, or the

. **

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...

: a special type of nonassociative algebra whose product satisfies the

. **

Jordan algebra In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms: # xy = yx (commutative law) # (xy)(xx) = x(y(xx)) (). The product of two elements ''x'' and ''y'' in a Jordan al ...

: a special type of nonassociative algebra whose product satisfies the

. *

Coalgebra In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagra ...

: a vector space with a "comultiplication" defined dually to that of associative algebras. * Lie coalgebra: a vector space with a "comultiplication" defined dually to that of Lie algebras. *

Graded algebra In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...

: 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 In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...

: an ''F'' vector space ''V'' with a

definite bilinear form In linguistics, definiteness is a semantic feature of noun phrases, distinguishing between referents or senses that are identifiable in a given context (definite noun phrases) and those which are not (indefinite noun phrases). The prototypical ...

. *

Bialgebra In mathematics, a bialgebra over a field ''K'' is a vector space over ''K'' which is both a unital associative algebra and a counital coassociative coalgebra. The algebraic and coalgebraic structures are made compatible with a few more axioms ...

: an associative algebra with a compatible coalgebra structure. *

Lie bialgebra In mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: it is a set with a Lie algebra and a Lie coalgebra structure which are compatible. It is a bialgebra where the multiplication is skew-symmetric and satisfies a dual Jacobi ...

: a Lie algebra with a compatible bialgebra structure. *

Hopf algebra Hopf is a German surname. Notable people with the surname include: * Eberhard Hopf (1902–1983), Austrian mathematician * Hans Hopf (1916–1993), German tenor * Heinz Hopf (1894–1971), German mathematician * Heinz Hopf (actor) (1934–2001), Sw ...

: a bialgebra with a connection axiom (antipode). *

Clifford algebra In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...

: an associative

\Z_2

-graded algebra additionally equipped with an exterior product from which several possible inner products may be derived.

Exterior algebra In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...

s and geometric algebras are special cases of this construction.

Algebraic structures with additional non-algebraic structure

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

Topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is al ...

s are vector spaces with a compatible

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...

. *

Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...

s: These are topological manifolds that also carry a compatible group structure. * Ordered groups, ordered rings and

ordered field In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered fie ...

s have algebraic structure compatible with an

order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...

on the set. *

Von Neumann algebra In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra. Von Neumann a ...

s: these are *-algebras on a

Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...

which are equipped with the

weak operator topology In functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space H, such that the functional sending an operator T to the complex number \langle Tx, y\rangle is ...

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 Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...

: *

s are used extensively in physics. A few well-known ones include the

orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...

s and the

unitary group In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group i ...

s. *

s *

Kac–Moody algebra In mathematics, a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently and simultaneously discovered them in 1968) is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a ...

*The

quaternions In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...

and more generally geometric algebras In

Mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...

: *

s are both rings and lattices, under their two operations. **

s are a special example of boolean algebras. *

Boundary algebra ''Laws of Form'' (hereinafter ''LoF'') is a book by G. Spencer-Brown, published in 1969, that straddles the boundary between mathematics and philosophy. ''LoF'' describes three distinct logical systems: * The "primary arithmetic" (described in Ch ...

MV-algebra In abstract algebra, a branch of pure mathematics, an MV-algebra is an algebraic structure with a binary operation \oplus, a unary operation \neg, and the constant 0, satisfying certain axioms. MV-algebras are the algebraic semantics of Łukasie ...

Computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includin ...

: *

Max-plus algebra In idempotent analysis, the tropical semiring is a semiring of extended real numbers with the operations of minimum (or maximum) and addition replacing the usual ("classical") operations of addition and multiplication, respectively. The tropi ...

Syntactic monoid In mathematics and computer science, the syntactic monoid M(L) of a formal language L is the smallest monoid that recognizes the language L. Syntactic quotient The free monoid on a given set is the monoid whose elements are all the strings of ...

* Transition monoid

References

Garrett Birkhoff Garrett Birkhoff (January 19, 1911 – November 22, 1996) was an American mathematician. He is best known for his work in lattice theory. The mathematician George Birkhoff (1884–1944) was his father. Life The son of the mathematician G ...

, 1967. ''Lattice Theory'', 3rd ed, AMS Colloquium Publications Vol. 25. American Mathematical Society. *———, and

Saunders MacLane Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician who co-founded category theory with Samuel Eilenberg. Early life and education Mac Lane was born in Norwich, Connecticut, near where his family lived in Taftville, ...

, 1999 (1967). ''Algebra'', 2nd ed. New York: Chelsea. *

George Boolos George Stephen Boolos (; 4 September 1940 – 27 May 1996) was an American philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology. Life Boolos is of Greek-Jewish descent. He graduated with an A.B. ...

and

Richard Jeffrey Richard Carl Jeffrey (August 5, 1926 – November 9, 2002) was an American philosopher, logician, and probability theorist. He is best known for developing and championing the philosophy of radical probabilism and the associated heuristic of ...

, 1980. ''Computability and Logic'', 2nd ed. Cambridge Univ. Press. *Dummit, David S., and Foote, Richard M., 2004. ''Abstract Algebra'', 3rd ed. John Wiley and Sons. *Grätzer, George, 1978. ''Universal Algebra'', 2nd ed. Springer. * David K. Lewis, 1991. ''Part of Classes''. Blackwell. * Michel, Anthony N., and Herget, Charles J., 1993 (1981). ''Applied Algebra and Functional Analysis''. Dover. *Potter, Michael, 2004. ''Set Theory and its Philosophy'', 2nd ed. Oxford Univ. Press. *Smorynski, Craig, 1991. ''Logical Number Theory I''. Springer-Verlag. A monograph available free online: * Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981.
A Course in Universal Algebra.
' Springer-Verlag. .

External links

* Jipsen: **Alphabetica
list
of algebra structures; includes many not mentioned here.
Online books and lecture notes.

containing about 50 structures, some of which do not appear above. Likewise, most of the structures above are absent from this map.
PlanetMath
topic index. *

Hazewinkel, Michiel Michiel Hazewinkel (born 22 June 1943) is a Dutch mathematician, and Emeritus Professor of Mathematics at the Centre for Mathematics and Computer Science and the University of Amsterdam, particularly known for his 1978 book ''Formal groups and ...

(2001)
Encyclopaedia of Mathematics.
' Springer-Verlag.

page on abstract algebra. *

Stanford Encyclopedia of Philosophy The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by Stanford University. E ...

Algebra
by Vaughan Pratt. {{Outline footer

algebraic structures In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...

Algebraic structures

Algebraic structures In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...