TheInfoList

OR:

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 ...
, an algebraic structure consists of a nonempty
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'' (called the underlying set, carrier set or domain), a collection of
operation Operation or Operations may refer to: Arts, entertainment and media * ''Operation'' (game), a battery-operated board game that challenges dexterity * Operation (music), a term used in musical set theory * ''Operations'' (magazine), Multi-Man ...
s on ''A'' (typically binary operations such as addition and multiplication), and a finite set of identities, known as axioms, that these operations must satisfy. An algebraic structure may be based on other algebraic structures with operations and axioms involving several structures. For instance, a
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 ...
involves a second structure called a field, and an operation called ''scalar multiplication'' between elements of the field (called ''
scalars 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 ...
''), and elements of the vector space (called '' vectors'').
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 ter ...
is the name that is commonly given to the study of algebraic structures. The general theory of algebraic structures has been formalized in
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 ...
.
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, ca ...
is another formalization that includes also other
mathematical structure In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additiona ...
s and functions between structures of the same type (
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
s). In universal algebra, an algebraic structure is called an ''algebra''; this term may be ambiguous, since, in other contexts, an algebra is an algebraic structure that is a
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 ...
over a field or a
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 ...
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 ...
. The collection of all structures of a given type (same operations and same laws) is called a variety in universal algebra; this term is also used with a completely different meaning in
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, as an abbreviation of
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
. In category theory, the collection of all structures of a given type and homomorphisms between them form a concrete category.

Introduction

Addition and multiplication are prototypical examples of operations that combine two elements of a set to produce a third element of the same set. These operations obey several algebraic laws. For example, and are associative laws, and and are commutative laws. Many systems studied by mathematicians have operations that obey some, but not necessarily all, of the laws of ordinary arithmetic. For example, the possible moves of an object in three-dimensional space can be combined by performing a first move of the object, and then a second move from its new position. Such moves, formally called rigid motions, obey the associative law, but fail to satisfy the commutative law. Sets with one or more operations that obey specific laws are called ''algebraic structures''. When a new problem involves the same laws as such an algebraic structure, all the results that have been proved using only the laws of the structure can be directly applied to the new problem. In full generality, algebraic structures may involve an arbitrary collection of operations, including operations that combine more than two elements (higher arity operations) and operations that take only one
argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialecti ...
( unary operations) or even zero arguments ( nullary operations). The examples listed below are by no means a complete list, but include the most common structures taught in undergraduate courses.

Common axioms

Equational axioms

An axiom of an algebraic structure often has the form of an
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), an ...
, that is, an
equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in F ...
such that the two sides of the
equals sign The equals sign (British English, Unicode) or equal sign (American English), also known as the equality sign, is the mathematical symbol , which is used to indicate equality in some well-defined sense. In an equation, it is placed between tw ...
are expressions that involve operations of the algebraic structure and variables. If the variables in the identity are replaced by arbitrary elements of the algebraic structure, the equality must remain true. Here are some common examples. ;
Commutativity 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 of ...
: An operation $*$ is ''commutative'' if $x*y=y*x$ for every and in the algebraic structure. ; Associativity: An operation $*$ is ''associative'' if $(x*y)*z=x*(y*z)$ for every , and in the algebraic structure. ; Left distributivity: An operation $*$ is ''left distributive'' with respect to another operation $+$ if $x*(y+z)=(x*y)+(x*z)$ for every , and in the algebraic structure (the second operation is denoted here as , because the second operation is addition in many common examples). ; Right distributivity: An operation $*$ is ''right distributive'' with respect to another operation $+$ if $(y+z)*x=(y*x)+(z*x)$ for every , and in the algebraic structure. ; Distributivity: An operation $*$ is ''distributive'' with respect to another operation $+$ if it is both left distributive and right distributive. If the operation $*$ is commutative, left and right distributivity are both equivalent to distributivity.

Existential axioms

Some common axioms contain an
existential clause An existential clause is a clause that refers to the existence or presence of something, such as "There is a God" and "There are boys in the yard". The use of such clauses can be considered analogous to existential quantification in predicate l ...
. In general, such a clause can be avoided by introducing further operations, and replacing the existential clause by an identity involving the new operation. More precisely, let us consider an axiom of the form ''"for all there is such that'' where is a -
tuple In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
of variables. Choosing a specific value of for each value of defines a function $\varphi:X\mapsto y,$ which can be viewed as an operation of arity , and the axiom becomes the identity $f\left(X,\varphi\left(X\right)\right)=g\left(X,\varphi\left(X\right)\right).$ The introduction of such auxiliary operation complicates slightly the statement of an axiom, but has some advantages. Given a specific algebraic structure, the proof that an existential axiom is satisfied consists generally of the definition of the auxiliary function, completed with straightforward verifications. Also, when computing in an algebraic structure, one generally uses explicitly the auxiliary operations. For example, in the case of
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual number ...
s, the additive inverse is provided by the unary minus operation $x\mapsto -x.$ Also, in
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 ...
, a variety is a class of algebraic structures that share the same operations, and the same axioms, with the condition that all axioms are identities. What precedes shows that existential axioms of the above form are accepted in the definition of a variety. Here are some of the most common existential axioms. ; Identity element :A binary operation $*$ has an identity element if there is an element such that $x*e=x\quad \text \quad e*x=x$ for all in the structure. Here, the auxiliary operation is the operation of arity zero that has as its result. ; Inverse element :Given a binary operation $*$ that has an identity element , an element is ''invertible'' if it has an inverse element, that is, if there exists an element $\operatorname\left(x\right)$ such that $\operatorname(x)*x=e \quad \text \quad x*\operatorname(x)=e.$For example, a group is an algebraic structure with a binary operation that is associative, has an identity element, and for which all elements are invertible.

Non-equational axioms

The axioms of an algebraic structure can be any first-order formula, that is a formula involving
logical connective In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary ...
s (such as ''"and"'', ''"or"'' and ''"not"''), and logical quantifiers ($\forall, \exists$) that apply to elements (not to subsets) of the structure. Such a typical axiom is inversion in fields. This axiom cannot be reduced to axioms of preceding types. (it follows that fields do not form a variety in the sense of
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 ...
.) It can be stated: ''"Every nonzero element of a field is invertible;"'' or, equivalently: ''the structure has a unary operation such that :$\forall x, \quad x=0 \quad\text \quad x \cdot\operatorname\left(x\right)=1.$ The operation can be viewed either as a partial operation that is not defined for ; or as an ordinary function whose value at 0 is arbitrary and must not be used.

Common algebraic structures

One set with operations

Simple structures: no binary operation: *
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. Group-like structures: one binary operation. The binary operation can be indicated by any symbol, or with no symbol (juxtaposition) as is done for ordinary multiplication of real numbers. * Group: a
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. 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 comm ...
: a group whose binary operation is commutative. Ring-like structures or Ringoids: two binary operations, often called addition and multiplication, with multiplication distributing over addition. * Ring: a semiring whose additive monoid is an abelian group. * Division ring: a nontrivial ring in which division by nonzero elements is defined. *
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 ...
: a ring in which the multiplication operation is commutative. * Field: a commutative division ring (i.e. a commutative ring which contains a multiplicative inverse for every nonzero element). Lattice structures: two or more binary operations, including 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 ...
, connected by the absorption law.Ringoids and lattices can be clearly distinguished despite both having two defining binary operations. In the case of ringoids, the two operations are linked by the distributive law; in the case of lattices, they are linked by the absorption law. Ringoids also tend to have numerical models, while lattices tend to have set-theoretic models. *
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 exist. * Bounded lattice: a lattice with a greatest element and least element. * Distributive lattice: a lattice in which each of meet and join distributes over the other. A power set under union and intersection forms a distributive lattice. *
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 i ...
: a complemented distributive lattice. Either of meet or join can be defined in terms of the other and complementation.

Two sets with operations

*
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 scalars, and the binary operation of ''scalar multiplication'' is a function ''R'' × ''M'' → ''M'', which satisfies several axioms. Counting the ring operations these systems have at least three operations. *
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 ...
: a module where the ring ''R'' is a division ring or field. *
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 ...
: a module over a field, which also carries a multiplication operation that is compatible with the module structure. This includes distributivity over addition and linearity with respect to multiplication. *
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 .

Hybrid structures

Algebraic structures can also coexist with added structure of non-algebraic nature, such as
partial order In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
or a
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 ...
. The added structure must be compatible, in some sense, with the algebraic structure. *
Topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
: a group with a topology compatible with the group operation. *
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 addi ...
: a topological group with a compatible smooth
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
structure. * Ordered groups, ordered rings and ordered fields: each type of structure with a compatible
partial order In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
. * Archimedean group: a linearly ordered group for which the Archimedean property holds. * Topological vector space: a vector space whose ''M'' has a compatible topology. * Normed vector space: a vector space with a compatible norm. If such a space is
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
(as a metric space) then it is called a Banach space. *
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 ...
: an inner product space over the real or complex numbers whose inner product gives rise to a Banach space structure. *
Vertex operator algebra In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory. In addition to physical applications, vertex operator algebras have proven usefu ...
* Von Neumann algebra: a *-algebra of operators on a Hilbert space 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 ...
.

Universal algebra

Algebraic structures are defined through different configurations of axioms.
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 ...
abstractly studies such objects. One major dichotomy is between structures that are axiomatized entirely by ''identities'' and structures that are not. If all axioms defining a class of algebras are identities, then this class is a variety (not to be confused with algebraic varieties of
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
). Identities are equations formulated using only the operations the structure allows, and variables that are tacitly
universally quantified In mathematical logic, a universal quantification is a type of Quantification (logic), quantifier, a logical constant which is interpretation (logic), interpreted as "given any" or "for all". It expresses that a predicate (mathematical logic), pr ...
over the relevant
universe The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the univers ...
. Identities contain no connectives, existentially quantified variables, or relations of any kind other than the allowed operations. The study of varieties is an important part of
universal algebra 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 ...
. An algebraic structure in a variety may be understood as the quotient algebra of term algebra (also called "absolutely free algebra") divided by the equivalence relations generated by a set of identities. So, a collection of functions with given signatures generate a free algebra, the term algebra ''T''. Given a set of equational identities (the axioms), one may consider their symmetric, transitive closure ''E''. The quotient algebra ''T''/''E'' is then the algebraic structure or variety. Thus, for example, groups have a signature containing two operators: the multiplication operator ''m'', taking two arguments, and the inverse operator ''i'', taking one argument, and the identity element ''e'', a constant, which may be considered an operator that takes zero arguments. Given a (countable) set of variables ''x'', ''y'', ''z'', etc. the term algebra is the collection of all possible terms involving ''m'', ''i'', ''e'' and the variables; so for example, ''m''(''i''(''x''), ''m''(''x'', ''m''(''y'',''e''))) would be an element of the term algebra. One of the axioms defining a group is the identity ''m''(''x'', ''i''(''x'')) = ''e''; another is ''m''(''x'',''e'') = ''x''. The axioms can be represented a
trees
These equations induce equivalence classes on the free algebra; the quotient algebra then has the algebraic structure of a group. Some structures do not form varieties, because either: # It is necessary that 0 ≠ 1, 0 being the additive identity element and 1 being a multiplicative identity element, but this is a nonidentity; # Structures such as fields have some axioms that hold only for nonzero members of ''S''. For an algebraic structure to be a variety, its operations must be defined for ''all'' members of ''S''; there can be no partial operations. Structures whose axioms unavoidably include nonidentities are among the most important ones in mathematics, e.g., fields and division rings. Structures with nonidentities present challenges varieties do not. For example, the
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
of two fields is not a field, because $\left(1,0\right)\cdot\left(0,1\right)=\left(0,0\right)$, but fields do not have
zero divisor In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right ze ...
s.

Category theory

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, ca ...
is another tool for studying algebraic structures (see, for example, Mac Lane 1998). A category is a collection of ''objects'' with associated ''morphisms.'' Every algebraic structure has its own notion of
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
, namely any function compatible with the operation(s) defining the structure. In this way, every algebraic structure gives rise to a category. For example, the category of groups has all
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
as objects and all group homomorphisms as morphisms. This concrete category may be seen as a category of sets with added category-theoretic structure. Likewise, the category of
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
s (whose morphisms are the continuous group homomorphisms) is a category of topological spaces with extra structure. A forgetful functor between categories of algebraic structures "forgets" a part of a structure. There are various concepts in category theory that try to capture the algebraic character of a context, for instance * algebraic category * essentially algebraic category * presentable category *
locally presentable category The theory of accessible categories is a part of mathematics, specifically of category theory. It attempts to describe categories in terms of the "size" (a cardinal number) of the operations needed to generate their objects. The theory originates ...
* monadic functors and categories *
universal property In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
.

Different meanings of "structure"

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

* Free object *
Mathematical structure In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additiona ...
* Signature (logic) * Structure (mathematical logic)

References

* * * ; Category theory * *