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| Algebraic structures |
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Inmathematics, analgebraic structure oralgebraic system[1] consists of a nonemptysetA (called theunderlying set,carrier set ordomain), a collection ofoperations onA (typicallybinary operations such 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 space involves a second structure called afield, and an operation calledscalar multiplication between elements of the field (calledscalars), and elements of the vector space (calledvectors).
Abstract algebra is the name that is commonly given to the study of algebraic structures. The general theory of algebraic structures has been formalized inuniversal algebra.Category theory is another formalization that includes othermathematical structures andfunctions between structures of the same type (homomorphisms).
In universal algebra, an algebraic structure is called analgebra;[2] this term may be ambiguous, since, in other contexts,an algebra is an algebraic structure that is a vector space over afield or amodule over acommutative ring.
The collection of all structures of a given type (same operations and same laws) is called avariety in 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.
Addition andmultiplication are prototypical examples ofoperations that 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 anda(bc) = (ab)c areassociative laws, anda +b =b +a andab =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 (higherarity operations) 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.
An axiom of an algebraic structure often has the form of anidentity, that is, anequation such that the two sides of theequals sign areexpressions that 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.
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 allX there isy such that", whereX is ak-tuple of variables. Choosing a specific value ofy for each value ofX defines a function which can be viewed as an operation ofarityk, and the axiom becomes the identity
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 inverse is provided by the unary minus operation
Also, inuniversal algebra, avariety 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.
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 () 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 avariety in the sense ofuniversal algebra.) It can be stated:"Every nonzero element of a field isinvertible;" or, equivalently:the structure has aunary operationinv such that
The operationinv can be viewed either as apartial operation that is not defined forx = 0; or as an ordinary function whose value at 0 is arbitrary and must not be used.
Simple structures:nobinary operation:
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.
Ring-like structures orRingoids:two binary operations, often calledaddition andmultiplication, with multiplicationdistributing over addition.
Lattice structures:two or more binary operations, including operations calledmeet and join, connected by theabsorption law.[3]
Algebraic structures can also coexist with added structure of non-algebraic nature, such aspartial order or atopology. The added structure must be compatible, in some sense, with the algebraic structure.
Algebraic structures are defined through different configurations ofaxioms.Universal algebra abstractly studies such objects. One major dichotomy is between structures that are axiomatized entirely byidentities and structures that are not. If all axioms defining a class of algebras are identities, then this class is avariety (not to be confused withalgebraic varieties ofalgebraic geometry).
Identities are equations formulated using only the operations the structure allows, and variables that are tacitlyuniversally quantified over the relevantuniverse. Identities contain noconnectives,existentially quantified variables, orrelations of 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 algebra of term algebra (also called "absolutelyfree algebra") divided by the equivalence relations generated by a set of identities. So, a collection of functions with givensignatures generate a free algebra, theterm algebraT. Given a set of equational identities (the axioms), one may consider their symmetric, transitive closureE. The quotient algebraT/E is 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 possibleterms involvingm,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 identitym(x,i(x)) =e; another ism(x,e) =x. The axioms can be represented astrees. These equations induceequivalence classes on the free algebra; the quotient algebra then has the algebraic structure of a group.
Some structures do not form varieties, because either:
Structures whose axioms unavoidably include nonidentities are among the most important ones in mathematics, e.g.,fields anddivision rings. Structures with nonidentities present challenges that varieties do not. For example, thedirect product of twofields is not a field, because, but fields do not havezero divisors.
Category theory is another tool for studying algebraic structures (see, for example, Mac Lane 1998). A category is a collection ofobjects with associatedmorphisms. Every algebraic structure has its own notion ofhomomorphism, namely anyfunction compatible with the operation(s) defining the structure. In this way, every algebraic structure gives rise to acategory. For example, thecategory of groups has allgroups as objects and allgroup homomorphisms as morphisms. Thisconcrete category may be seen as acategory of sets with added category-theoretic structure. Likewise, the category oftopological groups (whose morphisms are the continuous group homomorphisms) is acategory of topological spaces with extra structure. Aforgetful 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
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 ringstructure on the set", means that we have definedringoperations on the set. For another example, the group can be seen as a set that is equipped with analgebraic structure, namely theoperation.
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