mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 of ...

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 term ''a ...

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

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

s 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 on ...

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 or f ...

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 groups as the object of study, ...

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 form ...

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 can ...

s and

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

s 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 matrices. ...

. 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 groups as the object of study, ...

studies algebraic structures abstractly, rather than specific types of structures. *** Varieties **

Category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...

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 points ...

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 Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

: 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: 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 f ...

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

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 commut ...

: a group whose binary operation is commutative. * Quasigroup: 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, an ...

: a quasigroup with

identity Identity may refer to: * Identity document * Identity (philosophy) * Identity (social science) * Identity (mathematics) Arts and entertainment Film and television * ''Identity'' (1987 film), an Iranian film * ''Identity'' (2003 film), ...

. * Semilattice: a semigroup whose operation is idempotent and commutative. The binary operation can be called either

meet Meet may refer to: People with the name * Janek Meet (born 1974), Estonian footballer * Meet Mukhi (born 2005), Indian child actor Arts, entertainment, and media * ''Meet'' (TV series), an early Australian television series which aired on ABC du ...

join Join may refer to: * Join (law), to include additional counts or additional defendants on an indictment *In mathematics: ** Join (mathematics), a least upper bound of sets orders in lattice theory ** Join (topology), an operation combining two topo ...

. 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 the case of lattices, they are linked by the

absorption law In algebra, the absorption law or absorption identity is an identity linking a pair of binary operations. Two binary operations, ¤ and ⁂, are said to be connected by the absorption law if: :''a'' ¤ (''a'' ⁂ ''b'') = ''a'' ⁂ (''a'' ¤ ''b ...

. 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 Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...

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 additi ...

, with multiplication linked to addition by the distributive law. * 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 0 is an absorbing element in the sense that 0 ''x'' = 0 for all ''x''. * Near-ring: a semiring whose additive monoid is a (not necessarily abelian) group. *

Ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

: a semiring whose additive monoid is an abelian group. **

Commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...

: a ring in which the multiplication operation is commutative. ** Division ring: a nontrivial ring in which division by nonzero elements is defined. ** Integral domain: A nontrivial commutative ring in which the product of any two nonzero elements is nonzero. **

Field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

: a commutative division ring (i.e. a commutative ring which contains a multiplicative inverse for every nonzero element). *

Nonassociative ring 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'' i ...

s: 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 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 al ...

* Boolean ring: a commutative ring with idempotent multiplication operation. *

Kleene algebra In mathematics, a Kleene algebra ( ; named after Stephen Cole Kleene) is an idempotent (and thus partially ordered) semiring endowed with a closure operator. It generalizes the operations known from regular expressions. Definition Various ineq ...

s: a semiring with idempotent addition and a unary operation, the Kleene star, 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, 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 contrapositiv ...

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 u ...

. Robinson arithmetic with an axiom schema of

induction Induction, Inducible or Inductive may refer to: Biology and medicine * Labor induction (birth/pregnancy) * Induction chemotherapy, in medicine * Induced stem cells, stem cells derived from somatic, reproductive, pluripotent or other cell t ...

. 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 In mathematics, specifically order theory, the join of a subset S of a partially ordered set P is the supremum (least upper bound) of S, denoted \bigvee S, and similarly, the meet of S is the infimum (greatest lower bound), denoted \bigwedge S ...

, connected by the

. * Latticoid: meet and join

commute Commute, commutation or commutative may refer to: * Commuting, the process of travelling between a place of residence and a place of work Mathematics * Commutative property, a property of a mathematical operation whose result is insensitive to th ...

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.'' ...

: a lattice in which arbitrary

s 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 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 In the branch of mathematics called order theory, a modular lattice is a lattice (order), lattice that satisfies the following self-duality (order theory), dual condition, ;Modular law: implies where are arbitrary elements in the lattice, &nbs ...

: 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 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 in e ...

: 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 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 int ...

→, 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 Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modul ...

: an abelian group ''M'' and a ring ''R'' acting as operators on ''M''. The members of ''R'' are sometimes called

scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...

s, 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 field in t ...

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 ...

s, injective modules and flat modules 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 can ...

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

. **

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 th ...

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 more ...

decomposition into subspaces or "grades". **

Quadratic space 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 ...

: 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 a ...

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 (also ''R-algebra''): a module over a

commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...

''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''. **

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 addition ...

: 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 In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...

: 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 Jacobi identity, or the

. **

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

: a special type of nonassociative algebra whose product satisfies the Jacobi identity. **

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

Lie coalgebra In mathematics a Lie coalgebra is the dual structure to a Lie algebra. In finite dimensions, these are dual objects: the dual vector space to a Lie algebra naturally has the structure of a Lie coalgebra, and conversely. Definition Let ''E'' be ...

: 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 se ...

: 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 den ...

: an ''F'' vector space ''V'' with a definite bilinear form . *

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: 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), Swedis ...

: 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 algebras and

geometric algebra In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the ge ...

s 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 spaces are vector spaces with a compatible

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

. *

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 additio ...

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

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 r ...

: *

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 (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...

s and the unitary groups. *

s *

s * Kac–Moody algebra *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

s In

Mathematical logic Mathematical logic is the study of logic, 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 for ...

: *

s are both rings and lattices, under their two operations. ** Heyting algebras 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 Applied science, practical discipli ...

: *

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 tropical s ...

* Syntactic monoid * Transition monoid

References

* Garrett Birkhoff, 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. in ...

and Richard Jeffrey, 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 David Kellogg Lewis (September 28, 1941 – October 14, 2001) was an American philosopher who is widely regarded as one of the most important philosophers of the 20th century. Lewis taught briefly at UCLA and then at Princeton University fr ...

, 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. Eac ...

Algebra
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Vaughan Pratt Vaughan Pratt (born April 12, 1944) is a Professor Emeritus at Stanford University, who was an early pioneer in the field of computer science. Since 1969, Pratt has made several contributions to foundational areas such as search algorithms, sorti ...

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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 ...