Algebraic structure

Inmathematics,analgebraic structureconsists of a nonemptysetA(called theunderlying set,carrier setordomain), a collection ofoperationsonA(typicallybinary operationssuch as addition and multiplication), and a finite set ofidentities,known asaxioms,that these operations must satisfy.

An algebraic structure may be based on other algebraic structures with operations and axioms involving several structures. For instance, avector spaceinvolves a second structure called afield,and an operation calledscalar multiplicationbetween elements of the field (calledscalars), and elements of the vector space (calledvectors).

Abstract algebrais the name that is commonly given to the study of algebraic structures. The general theory of algebraic structures has been formalized inuniversal algebra.Category theoryis another formalization that includes also othermathematical structuresandfunctionsbetween structures of the same type (homomorphisms).

In universal algebra, an algebraic structure is called analgebra;^[1]this term may be ambiguous, since, in other contexts,an algebrais an algebraic structure that is avector spaceover afieldor amoduleover acommutative ring.

The collection of all structures of a given type (same operations and same laws) is called avarietyin universal algebra; this term is also used with a completely different meaning inalgebraic geometry,as an abbreviation ofalgebraic variety.In category theory, the collection of all structures of a given type and homomorphisms between them form aconcrete category.

Introduction[edit]

Additionandmultiplicationare prototypical examples ofoperationsthat combine two elements of a set to produce a third element of the same set. These operations obey several algebraic laws. For example, $a + (b + c) = (a + b) + c$ and $a (bc) = (ab) c$ areassociative laws,and $a + b = b + a$ and $ab = ba$ arecommutative 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 calledrigid motions,obey the associative law, but fail to satisfy the commutative law.

Sets with one or more operations that obey specific laws are calledalgebraic 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 (higherarityoperations) and operations that take only oneargument(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[edit]

Equational axioms[edit]

An axiom of an algebraic structure often has the form of anidentity,that is, anequationsuch that the two sides of theequals signareexpressionsthat involve operations of the algebraic structure andvariables.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 $*$ iscommutativeif $x*y=y*x$ for every $x$ and $y$ in the algebraic structure.
Associativity: An operation $*$ isassociativeif $(x*y)*z=x*(y*z)$ for every $x$ , $y$ and $z$ in the algebraic structure.
Left distributivity: An operation $*$ isleft distributivewith respect to another operation $+$ if $x*(y+z)=(x*y)+(x*z)$ for every $x$ , $y$ and $z$ 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 $*$ isright distributivewith respect to another operation $+$ if $(y+z)*x=(y*x)+(z*x)$ for every $x$ , $y$ and $z$ in the algebraic structure.
Distributivity: An operation $*$ isdistributivewith 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[edit]

Some common axioms contain anexistential 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 $X$ there is $y$ such that $f(X,y)=g(X,y)$ ",where $X$ is a $k$ -tupleof variables. Choosing a specific value of $y$ for each value of $X$ defines a function $\varphi:X\mapsto y,$ which can be viewed as an operation ofarity $k$ ,and the axiom becomes the identity $f(X,\varphi (X))=g(X,\varphi (X)).$

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 ofnumbers,theadditive inverseis provided by the unary minus operation $x\mapsto -x.$

Also, inuniversal algebra,avarietyis 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: Abinary operation $*$ has an identity element if there is an element $e$ such that $x*e=x\quad {\text{and}}\quad e*x=x$ for all $x$ in the structure. Here, the auxiliary operation is the operation of arity zero that has $e$ as its result.
Inverse element: Given a binary operation $*$ that has an identity element $e$ ,an element $x$ isinvertibleif it has an inverse element, that is, if there exists an element $\operatorname {inv} (x)$ such that $\operatorname {inv} (x)*x=e\quad {\text{and}}\quad x*\operatorname {inv} (x)=e.$ For example, agroupis an algebraic structure with a binary operation that is associative, has an identity element, and for which all elements are invertible.

Non-equational axioms[edit]

The axioms of an algebraic structure can be anyfirst-order formula,that is a formula involvinglogical connectives(such as"and","or"and"not"), andlogical quantifiers( $\forall,\exists$ ) that apply to elements (not to subsets) of the structure.

Such a typical axiom is inversion infields.This axiom cannot be reduced to axioms of preceding types. (it follows that fields do not form avarietyin the sense ofuniversal algebra.) It can be stated:"Every nonzero element of a field isinvertible;"or, equivalently:the structure has aunary operation $inv$ such that

\forall x,\quad x=0\quad {\text{or}}\quad x\cdot \operatorname {inv} (x)=1.

The operation $inv$ can be viewed either as apartial operationthat is not defined for $x = 0$ ;or as an ordinary function whose value at 0 is arbitrary and must not be used.

Common algebraic structures[edit]

One set with operations[edit]

Simple structures:nobinary operation:

Set:a degenerate algebraic structureShaving no operations.

Group-like structures:onebinary 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:amonoidwith a unary operation (inverse), giving rise toinverse elements.
Abelian group:a group whose binary operation iscommutative.

Ring-like structuresorRingoids:twobinary operations, often calledadditionandmultiplication,with multiplicationdistributingover addition.

Ring:a semiring whose additive monoid is an abelian group.
Division ring:anontrivialring in whichdivisionby 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:twoor more binary operations, including operations calledmeet and join,connected by theabsorption law.^[2]

Complete lattice:a lattice in which arbitrarymeet and joinsexist.
Bounded lattice:a lattice with agreatest elementand least element.
Distributive lattice:a lattice in which each of meet and joindistributesover the other. Apower setunder union and intersection forms a distributive lattice.
Boolean algebra:a complemented distributive lattice. Either of meet or join can be defined in terms of the other and complementation.

Two sets with operations[edit]

Module:an abelian groupMand a ringRacting as operators onM.The members ofRare sometimes calledscalars,and the binary operation ofscalar multiplicationis a functionR×M→M,which satisfies several axioms. Counting the ring operations these systems have at least three operations.
Vector space:a module where the ringRis adivision ringorfield.

Algebra over a field:a module over a field, which also carries a multiplication operation that is compatible with the module structure. This includes distributivity over addition andlinearitywith respect to multiplication.
Inner product space:anFvector spaceVwith adefinite bilinear formV×V→F.

Hybrid structures[edit]

Algebraic structures can also coexist with added structure of non-algebraic nature, such aspartial orderor atopology.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:a topological group with a compatible smoothmanifoldstructure.
Ordered groups,ordered ringsandordered fields:each type of structure with a compatiblepartial order.
Archimedean group:a linearly ordered group for which theArchimedean propertyholds.
Topological vector space:a vector space whoseMhas a compatible topology.
Normed vector space:a vector space with a compatiblenorm.If such a space iscomplete(as a metric space) then it is called aBanach space.
Hilbert space: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 theweak operator topology.

Universal algebra[edit]

Algebraic structures are defined through different configurations ofaxioms.Universal algebraabstractly studies such objects. One major dichotomy is between structures that are axiomatized entirely byidentitiesand structures that are not. If all axioms defining a class of algebras are identities, then this class is avariety(not to be confused withalgebraic varietiesofalgebraic geometry).

Identities are equations formulated using only the operations the structure allows, and variables that are tacitlyuniversally quantifiedover the relevantuniverse.Identities contain noconnectives,existentially quantified variables,orrelationsof any kind other than the allowed operations. The study of varieties is an important part ofuniversal algebra.An algebraic structure in a variety may be understood as thequotient algebraof term algebra (also called "absolutelyfree algebra") divided by the equivalence relations generated by a set of identities. So, a collection of functions with givensignaturesgenerate a free algebra, theterm algebraT.Given a set of equational identities (the axioms), one may consider their symmetric, transitive closureE.The quotient algebraT/Eis then the algebraic structure or variety. Thus, for example, groups have a signature containing two operators: the multiplication operatorm,taking two arguments, and the inverse operatori,taking one argument, and the identity elemente,a constant, which may be considered an operator that takes zero arguments. Given a (countable) set of variablesx,y,z,etc. the term algebra is the collection of all possibletermsinvolvingm,i,eand 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 identitym(x,i(x)) =e;another ism(x,e) =x.The axioms can be represented astrees.These equations induceequivalence classeson 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 additiveidentity elementand 1 being a multiplicative identity element, but this is a nonidentity;
Structures such as fields have some axioms that hold only for nonzero members ofS.For an algebraic structure to be a variety, its operations must be defined forallmembers ofS;there can be no partial operations.

Structures whose axioms unavoidably include nonidentities are among the most important ones in mathematics, e.g.,fieldsanddivision rings.Structures with nonidentities present challenges varieties do not. For example, thedirect productof twofieldsis not a field, because $(1,0)\cdot (0,1)=(0,0)$ ,but fields do not havezero divisors.

Category theory[edit]

Category theoryis another tool for studying algebraic structures (see, for example, Mac Lane 1998). A category is a collection ofobjectswith associatedmorphisms.Every algebraic structure has its own notion ofhomomorphism,namely anyfunctioncompatible with the operation(s) defining the structure. In this way, every algebraic structure gives rise to acategory.For example, thecategory of groupshas allgroupsas objects and allgroup homomorphismsas morphisms. Thisconcrete categorymay be seen as acategory of setswith added category-theoretic structure. Likewise, the category oftopological groups(whose morphisms are the continuous group homomorphisms) is acategory of topological spaceswith extra structure. Aforgetful functorbetween 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

Different meanings of "structure"[edit]

In a slightabuse 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 ringstructureon the set $A$ ",means that we have definedringoperationson the set $A$ .For another example, the group $(\mathbb {Z},+)$ can be seen as a set $\mathbb {Z}$ that is equipped with analgebraic structure,namely theoperation $+$ .

Notes[edit]

^P.M. Cohn. (1981)Universal Algebra,Springer, p. 41.
^Ringoids andlatticescan be clearly distinguished despite both having two defining binary operations. In the case of ringoids, the two operations are linked by thedistributive law;in the case of lattices, they are linked by theabsorption law.Ringoids also tend to have numericalmodels,while lattices tend to haveset-theoreticmodels.

References[edit]

Mac Lane, Saunders;Birkhoff, Garrett(1999),Algebra(2nd ed.), AMS Chelsea,ISBN 978-0-8218-1646-2
Michel, Anthony N.; Herget, Charles J. (1993),Applied Algebra and Functional Analysis,New York:Dover Publications,ISBN 978-0-486-67598-5
Burris, Stanley N.; Sankappanavar, H. P. (1981),A Course in Universal Algebra,Berlin, New York:Springer-Verlag,ISBN 978-3-540-90578-3

Category theory

Mac Lane, Saunders(1998),Categories for the Working Mathematician(2nd ed.), Berlin, New York: Springer-Verlag,ISBN 978-0-387-98403-2
Taylor, Paul (1999),Practical foundations of mathematics,Cambridge University Press,ISBN 978-0-521-63107-5

External links[edit]

Jipsen's algebra structures.Includes many structures not mentioned here.
Mathworldpage on abstract algebra.
Stanford Encyclopedia of Philosophy:AlgebrabyVaughan Pratt.

[1] P.M. Cohn. (1981)Universal Algebra,Springer, p. 41.

[2] Ringoids andlatticescan be clearly distinguished despite both having two defining binary operations. In the case of ringoids, the two operations are linked by thedistributive law;in the case of lattices, they are linked by theabsorption law.Ringoids also tend to have numericalmodels,while lattices tend to haveset-theoreticmodels.

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v t e Algebra
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Basic concepts	Algebraic expression Equation(Linear equation,Quadratic equation) Function(Polynomial function) Inequality(Linear inequality) Operation(Addition,Multiplication) Relation(Equivalence relation) Variable
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