In
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
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 ...
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
Jipsenan
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:
**
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 ...
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
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. 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
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 ...
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 ...
or
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
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 ...
is
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
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 ...
.
*
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
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 ...
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 ...
s,
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
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 ...
.
**
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
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 ...
.
*
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
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 ...
.
**
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
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 ...
.
*
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
-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 ...
:
*
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 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.
*
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 ...
s
*
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 ...
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
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
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 ...
:
*
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 ...
s are both rings and lattices, under their two operations.
**
Heyting algebras are a special example of boolean algebras.
*
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 ...
*
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 ...
In
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
See also
*
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 ...
**
Outline of abstract algebra
*
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, ...
**
Variety (universal algebra)
*
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.
...
**
Outline of linear algebra
This is an outline of topics related to linear algebra, the branch of mathematics concerning linear equations and linear maps and their representations in vector spaces and through matrices.
Linear equations
Linear equation
* System of linear eq ...
*
Arity
Arity () is the number of arguments or operands taken by a function, operation or relation in logic, mathematics, and computer science. In mathematics, arity may also be named ''rank'', but this word can have many other meanings in mathematics. In ...
*
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 ...
*
Free object
*
Operation (mathematics)
In mathematics, an operation is a function which takes zero or more input values (also called "''operands''" or "arguments") to a well-defined output value. The number of operands is the arity of the operation.
The most commonly studied operatio ...
*
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 ...
*
First-order theories
In first-order logic, a first-order theory is given by a set of axioms in some
language. This entry lists some of the more common examples used in model theory and some of their properties.
Preliminaries
For every natural mathematical structure ...
*
Mathematical lists
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
listof 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.
PlanetMathtopic 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 ...
Algebraby
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
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 ...