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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 ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically
binary operation 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 ...
s such as addition and multiplication), and a finite set of identities, known as
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 Ancient Greek includes the forms of the ...
s, 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 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 th ...
involves a second structure called a field, and an operation called ''scalar multiplication'' between elements of the field (called '' scalars''), and elements of the vector space (called '' vectors'').
Abstract algebra 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 i ...
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.
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 Algebraic ...
is another formalization that includes also other mathematical structures and functions between structures of the same type (
homomorphism In algebra Algebra () is one of the broad areas of 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 quantitie ...
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 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 th ...
over a field or a module over a commutative ring. 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 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. ...
, as an abbreviation of
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry Algebraic geometry is a branch of mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes ...
. In category theory, the collection of all structures of a given type and homomorphisms between them form a concrete category.

# Introduction

Addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' ...
and
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations ( c ...
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 premise A premise or premiss is a true or false statement that helps form the body of an argument, which logically leads to a true or false conclusion. A premise makes a declarative sta ...
(
unary operation 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 ...
s) 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, that is, an
equation 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 ...
such that the two sides of the
equals sign The equals sign (British English British English (BrE, en-GB, or BE) is, according to Oxford Dictionaries, " English as used in Great Britain Great Britain is an island in the North Atlantic Ocean The Atlantic Ocean is ...
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: An operation $*$ is ''commutative'' if $x*y=y*x$ for every and in the algebraic structure. ;
Associativity 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 i ...
: 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. 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 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 m ...
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, 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 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 mod ...
:A
binary operation 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 ...
$*$ 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 A group is a 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 universa ...
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 Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions fol ...
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.) It can be stated: ''"Every nonzero element of a field is invertible;"'' or, equivalently: ''the structure has a
unary operation 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 ...
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 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 ...
: * Set: 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 group is a 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 universa ...
: a monoid with a unary operation (inverse), giving rise to inverse elements. *
Abelian group 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 m ...
: a group whose binary operation is
commutative 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 ...
. Ring-like structures or Ringoids: two binary operations, often called
addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' ...
and
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations ( c ...
, 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: 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, 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
model A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models ...
s, while lattices tend to have set-theoretic models.
* Complete lattice: a lattice in which arbitrary meet and joins exist. *
Bounded lattice A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum In mathematics M ...
: a lattice with a
greatest element 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 ...
and least element. * Distributive lattice: a lattice in which each of meet and join distributes over the other. A
power set 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 ...
under union and intersection forms a distributive lattice. *
Boolean algebra 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 ...
: a complemented distributive lattice. Either of meet or join can be defined in terms of the other and complementation.

## Two sets with operations

* Module: an abelian group ''M'' and a ring ''R'' acting as operators on ''M''. The members of ''R'' are sometimes called scalars, and the binary operation of ''scalar multiplication'' is a function ''R'' × ''M'' → ''M'', which satisfies several axioms. Counting the ring operations these systems have at least three operations. *
Vector space In mathematics and 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 th ...
: a module where the ring ''R'' is a division ring or field. *
Algebra over a field 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 m ...
: 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 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 ...
: 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 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 m ...
or a
topology 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 i ...
. The added structure must be compatible, in some sense, with the algebraic structure. * Topological group: a group with a topology compatible with the group operation. *
Lie group 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 i ...
: a topological group with a compatible smooth
manifold 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 i ...
structure. * Ordered groups, ordered rings and ordered fields: each type of structure with a compatible
partial order 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 m ...
. * Archimedean group: a linearly ordered group for which the Archimedean property holds. *
Topological vector space 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 mo ...
: 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 (as a metric space) then it is called a
Banach space 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 mod ...
. *
Hilbert space 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 ...
: an inner product space over the real or complex numbers whose inner product gives rise to a Banach space structure. * Vertex operator algebra * Von Neumann algebra: a *-algebra of operators on a Hilbert space equipped with the weak operator topology.

# Universal algebra

Algebraic structures are defined through different configurations 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 Ancient Greek includes the forms of the ...
s. Universal algebra 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 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. ...
). Identities are equations formulated using only the operations the structure allows, and variables that are tacitly universally quantified 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 univer ...
. 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. An algebraic structure in a variety may be understood as the quotient algebra of term algebra (also called "absolutely
free algebra 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 mo ...
") 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 In universal algebra and mathematical logic Mathematical logic is the study of formal logic within mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the sp ...
''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 class 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 ...
es 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 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 mod ...
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 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 mod ...
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 divisors.

# 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 Algebraic ...
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 Algebra () is one of the broad areas of 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 quantitie ...
, namely any function compatible with the operation(s) defining the structure. In this way, every algebraic structure gives rise to a
category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce ...
. For example, the category of groups has all groups as objects and all
group homomorphism 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 ...
s as morphisms. This concrete category may be seen as a
category of sets In the mathematical field of 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 founda ...
with added category-theoretic structure. Likewise, the category of topological groups (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 * monadic functors and categories * universal property.

# 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 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 mod ...
* Mathematical structure * Signature (logic) * Structure (mathematical logic)

# References

* * * ; Category theory * *