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| {{Algebraic structures}}
| | You might discover convenient ways to speed up computer by making the most from the built in tools in a Windows plus obtaining the Service Pack updates-speed up your PC plus fix error. Simply follow a limited protocols to instantly create a computer quick than ever.<br><br>Before actually ordering the software it really is best to check found on the firms that make the software. If you could find details on the kind of reputation each firm has, maybe the risk of malicious programs is reduced. Software from reputed companies have aided me, plus various other users, to make my PC run quicker.. If the product description does not look superior to we, refuses to include details about the software, does not include the scan functions, you should go for another 1 that ensures you're paying for what you wish.<br><br>The error is basically a result of issue with Windows Installer package. The Windows Installer is a tool employed to install, uninstall plus repair the most programs on your computer. Let you discuss a limited things that helped a great deal of people that facing the similar matter.<br><br>Review your files plus clean it up frequently. Destroy all of the unnecessary plus unused files considering they just jam your computer system. It can surely better the speed of the computer plus be cautious which your computer do not infected by a virus. Remember usually to update the antivirus software every time. If you never use the computer pretty frequently, you are able to take a free antivirus.<br><br>Another common cause of PC slow down is a corrupt registry. The registry is a really important component of computers running on Windows platform. When this gets corrupted your PC will slowdown, or worse, not commence at all. Fixing the registry is easy with all the utilize of the system plus [http://bestregistrycleanerfix.com/regzooka zookaware].<br><br>Software errors or hardware mistakes which happen when running Windows and intermittent errors are the general reasons for a blue screen physical memory dump. New software or motorists which have been installed or changes inside the registry settings are the typical s/w causes. Intermittent errors refer to failed program memory/ difficult disk or over heated processor plus these too may result the blue screen physical memory dump error.<br><br>Most probably if you are experiencing a slow computer it can be a couple years old. We also will not have been told which while you utilize the computer everyday; there are certain details that it requires to continue running inside its best performance. You moreover will not even own any diagnostic tools that can receive a PC running like modern again. Well do not allow which stop you from getting your program cleaned. With access to the web we can find the tools that will help you get a system running like hot again.<br><br>A registry cleaner is a program that cleans the registry. The Windows registry usually gets flooded with junk information, information that has not been removed from uninstalled programs, erroneous file organization plus additional computer-misplaced entries. These neat little system software tools are very popular nowadays plus you can find many advantageous ones on the Internet. The good ones give you way to keep, clean, update, backup, and scan the System Registry. When it finds supposedly unwanted elements inside it, the registry cleaner lists them and recommends the user to delete or repair these orphaned entries and corrupt keys. |
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| In [[mathematics]], and more specifically in [[abstract algebra]], the term '''algebraic structure''' generally refers to an arbitrary [[Set (mathematics)|set]] (called '''carrier set''' or '''underlying set''') with one or more [[finitary operation]]s defined on it.<ref>P.M. Cohn. (1981) ''Universal Algebra'', Springer, p. 41.</ref>
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| Common examples of algebraic structures include [[group (mathematics)|groups]], [[ring (mathematics)|rings]], [[field (mathematics)|fields]], and [[lattice (order)|lattices]]. More complex algebraic structures can be defined by introducing multiple operations, different underlying sets, or by altering the defining axioms. Examples of more complex structures include [[vector space]]s, [[module (mathematics)|modules]], and [[algebra (ring theory)|algebras]].
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| The properties of specific algebraic structures are studied in the branch known as [[abstract algebra]]. The general theory of algebraic [[structure (mathematical logic)|structure]]s has been formalized in [[universal algebra]]. [[Category theory]] is used to study the relationships between two or more classes of algebraic structures, often of different kinds. For example, [[Galois theory]] studies the connection between certain ''fields'' and ''groups'', algebraic structures of two different kinds.
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| In a slight abuse of notation, the word "structure" can also refer only to the operations on a structure, and not the underlying set itself. For example, a phrase "we have defined a ring ''structure'' (a ''structure'' of ring) on the set <math>A</math>" means that we have defined [[ring (mathematics)|ring]] operations on the set <math>A</math>. For another example, the group <math>(\mathbb Z, +)</math> can be seen as a set <math>\mathbb Z</math> that is equipped with an ''algebraic structure,'' namely the operation <math>+</math>.
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| ==Introduction==
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| Addition and multiplication on numbers are the prototypical example of an operation that combines two elements of a set to produce a third. These operations obey several algebraic laws. For example ''a'' + (''b'' + ''c'') = (''a'' + ''b'') + ''c'' and ''a''(''bc'') = (''ab'')''c'', both example of the ''associative law''. Also ''a'' + ''b'' = ''b'' + ''a'', and ''ab'' = ''ba'', the ''commutative law.'' Many systems studied by mathematicians have operations that obey some, but not necessarily all, of the laws of ordinary arithmetic. For example, rotations of objects in three-dimensional space can be combined by performing the first rotation and then applying the second rotation to the object in its new orientation. This operation on rotations obeys the associative law, but can fail the commutative law.
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| Mathematicians give names to sets with one or more operations that obey a particular collection of laws, and study them in the abstract as algebraic structures. When a new problem can be shown to follow the laws of one of these algebraic structures, all the work that has been done on that category in the past can be applied to the new problem.
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| In full generality, algebraic structures may involve an arbitrary number of sets and operations that can combine more than two elements (higher [[arity]]), but this article focuses on binary operations on one or two sets. The examples here are by no means a complete list, but they are meant to be a representative list and include the most common structures. Longer lists of algebraic structures may be found in the external links and within ''[[:Category:Algebraic structures]].'' Structures are listed in approximate order of increasing complexity.
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| ==Examples==
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| ===One set with operations===
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| '''Simple structures''': '''No''' [[binary operation]]:
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| * [[Set (mathematics)|Set]]: a degenerate algebraic structure having no operations.
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| * [[Pointed set]]: ''S'' has one or more distinguished elements, often 0, 1, or both.
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| * Unary system: ''S'' and a single [[unary operation]] over ''S''.
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| * Pointed unary system: a unary system with ''S'' a pointed set.
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| '''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.
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| * [[Magma (algebra)|Magma or groupoid]]: ''S'' and a single binary operation over ''S''.
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| * [[Semigroup]]: an [[associative]] magma.
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| * [[Monoid]]: a semigroup with [[Identity element|identity]].
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| * [[Group (mathematics)|Group]]: a monoid with a unary operation (inverse), giving rise to [[inverse element]]s.
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| * [[Abelian group]]: a group whose binary operation is [[commutative]].
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| * [[Semilattice]]: a semigroup whose operation is [[idempotence|idempotent]] and commutative. The binary operation can be called either [[meet (mathematics)|meet]] or [[join (mathematics)|join]].
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| '''Ring-like structures''' or '''Ringoids''': '''Two''' binary operations, often called [[addition]] and [[multiplication]], with multiplication [[distributivity|distributing]] over addition.
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| * [[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 satisfies 0 ''x'' = 0 for all ''x''.
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| * [[Near-ring]]: a semiring whose additive monoid is a (not necessarily Abelian) group.
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| * [[Ring (mathematics)|Ring]]: a semiring whose additive monoid is an Abelian group.
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| * [[Lie ring]]: a ringoid whose additive monoid is an abelian group, but whose multiplicative operation satisfies the [[Jacobi identity]] rather than associativity.
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| * [[Boolean ring]]: a commutative ring with idempotent multiplication operation.
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| * [[field (mathematics)|Field]]: a commutative ring which contains a multiplicative inverse for every nonzero element
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| * [[Kleene algebra]]s: a semiring with idempotent addition and a unary operation, the [[Kleene star]], satisfying additional properties.
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| * [[*-algebra]]: a ring with an additional unary operation (*) satisfying additional properties.
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| '''Lattice structures''': '''Two''' or more binary operations, including operations called [[meet and join]], connected by the [[absorption law]].<ref>Ringoids and [[Lattice (order)|lattice]]s 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 theory|model]]s, while lattices tend to have [[set theory|set-theoretic]] models.
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| </ref> | |
| * [[Complete lattice]]: a lattice in which arbitrary [[meet and join]]s exist.
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| * [[Bounded lattice]]: a lattice with a [[greatest element]] and least element.
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| * [[Complemented lattice]]: a bounded lattice with a unary operation, complementation, denoted by [[reverse Polish notation|postfix]] <sup>[[⊥]]</sup><!-- or [[prefix notation|prefix]] [[¬]]-->. The join of an element with its complement is the greatest element, and the meet of the two elements is the least element.
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| * [[Modular lattice]]: a lattice whose elements satisfy the additional ''modular identity''.
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| * [[Distributive lattice]]: a lattice in which each of meet and join [[distributive lattice|distributes]] over the other. Distributive lattices are modular, but the converse does not hold.
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| * [[Boolean algebra (structure)|Boolean algebra]]: 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.
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| * [[Heyting algebra]]: a bounded distributive lattice with an added binary operation, [[relative pseudo-complement]], denoted by [[infix]] →, and governed by the axioms ''x'' → ''x'' = 1, ''x'' (''x'' → ''y'') = ''x y'', ''y'' (''x'' → ''y'') = ''y'', ''x'' → (''y z'') = (''x'' → ''y'') (''x'' → ''z'').
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| '''Arithmetics''': '''Two''' [[binary operation]]s, addition and multiplication. ''S'' is an [[infinite set]]. Arithmetics are pointed unary systems, whose [[unary operation]] is [[injective]] [[successor function|successor]], and with distinguished element 0.
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| *[[Robinson arithmetic]]. Addition and multiplication are [[Recursion|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.
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| * [[Peano arithmetic]]. Robinson arithmetic with an [[axiom schema]] of [[mathematical induction|induction]]. Most ring and field axioms bearing on the properties of addition and multiplication are theorems of Peano arithmetic or of proper extensions thereof.
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| ===Two sets with operations===
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| '''[[module (mathematics)|Module]]-like structures:''' composite systems involving two sets and employing at least two binary operations.
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| * [[Group with operators]]: a group ''G'' with a set Ω and a binary operation Ω × ''G'' → ''G'' satisfying certain axioms.
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| *[[module (mathematics)|Module]]: an Abelian group ''M'' and a ring ''R'' acting as operators on ''M''. The members of ''R'' are sometimes called [[scalar (mathematics)|scalar]]s, 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.
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| *[[Vector space]]: a module where the ring ''R'' is a [[division ring]] or [[field (mathematics)|field]].
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| *[[Graded vector space]]: a vector space with a [[Direct sum of modules|direct sum]] decomposition breaking the space into "grades".
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| *[[Quadratic space]]: a vector space ''V'' over a field ''F'' with a function from ''V'' into ''F'' satisfying certain properties. Every quadratic space is also an inner product space (see below).
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| '''[[algebra (ring theory)|Algebra]]-like structures''': composite system defined over two sets, a ring ''R'' and a ''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''.
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| *[[Algebra over a ring]] (also ''R-algebra''): a module over a [[commutative ring]] ''R'', which also carries a multiplication operation that is compatible with the module structure. This includes distributivity over addition and [[Bilinear map|linearity]] with respect to multiplication by elements of ''R''. The theory of an [[algebra over a field]] is especially well developed.
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| *[[Associative algebra]]: an algebra over a ring such that the multiplication is [[associative property|associative]].
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| *[[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 [[alternative algebra|alternation]], the [[Jacobi identity]], or the [[Jordan identity]].
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| *[[Coalgebra]]: a vector space with a "comultiplication" defined dually to that of associative algebras.
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| *[[Lie algebra]]: a special type of nonassociative algebra whose product satisfies the [[Jacobi identity]].
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| *[[Lie coalgebra]]: a vector space with a "comultiplication" defined dually to that of Lie algebras.
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| *[[Graded algebra]]: 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.
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| *[[Inner product space]]: an ''F'' vector space ''V'' with a bilinear binary operation from ''V'' × ''V'' → ''F''.
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| '''Three''' [[binary operation]]s:
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| *[[Quasigroup]]: may be represented as an algebraic structure with one or three binary operations.<ref>{{cite book
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| | title =An Introduction to Quasigroups and Their Representations
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| | author =Jonathan D. H. Smith
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| | publisher =Chapman & Hall
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| | url =http://books.google.co.uk/books?id=NfWlUZSOwSkC&printsec=frontcover&dq=quasigroups&source=bl&ots=8ZOf4xvSh6&sig=MyWk4X7vHJL3WkJtPq-Rq3NhLns&hl=en&sa=X&ei=F9caUMjEG4eb1AWAhYGQAw&redir_esc=y#v=onepage&q=quasigroups&f=false
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| | accessdate = 2012-08-02 }}</ref>
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| '''Four''' or more binary operations:
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| *[[Bialgebra]]: an associative algebra with a compatible coalgebra structure.
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| *[[Lie bialgebra]]: a Lie algebra with a compatible bialgebra structure.
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| *[[Clifford algebra]]: a graded associative algebra equipped with an [[exterior product]] from which may be derived several possible inner products. [[Exterior algebra]]s and [[geometric algebra]]s are special cases of this construction.
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| ==Hybrid structures==
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| Algebraic structures can also coexist with added structure of a non-algebraic nature, such as a partial order or a [[topology]]. The added structure must be compatible, in some sense, with the algebraic structure.
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| * [[Topological group]]: a group with a topology compatible with the group operation.
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| * [[Lie group]]: a topological group with a compatible smooth [[manifold]] structure.
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| * [[Ordered group]]s, [[ordered ring]]s and [[ordered field]]s: each type of structure with a compatible [[partial order]].
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| * [[Archimedean group]]: a linearly ordered group for which the [[Archimedean property]] holds.
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| * [[Topological vector space]]: a vector space whose ''M'' has a compatible topology.
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| * [[Normed vector space]]: a vector space with a compatible [[norm (mathematics)|norm]]. If such a space is [[complete metric space|complete]] (as a metric space) then it is called a [[Banach space]].
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| * [[Hilbert space]]: an inner product space over the real or complex numbers whose inner product gives rise to a Banach space structure.
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| * [[Vertex operator algebra]]
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| * [[Von Neumann algebra]]: a *-algebra of operators on a Hilbert space equipped with the [[weak operator topology]].
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| ==Universal algebra==
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| Algebraic structures are defined through different configurations of [[axiom]]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 the class of objects is a [[variety (universal algebra)|variety]] (not to be confused with [[algebraic variety]] in the sense of [[algebraic geometry]]).
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| Identities are equations formulated using only the operations the structure allows, and variables that are tacitly [[universal quantifier|universally quantified]] over the relevant [[universe (mathematics)|universe]]. Identities contain no [[Logical connective|connectives]], [[quantification|existentially quantified variables]], or [[finitary relation|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 object|free algebra]]") divided by the equivalence relations generated by a set of identities. So, a collection of functions with given [[signature (logic)|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 [[term (mathematics)|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 as [http://ncatlab.org/nlab/show/variety+of+algebras#examples_4 trees]. These equations induce [[equivalence class]]es on the free algebra; the quotient algebra then has the algebraic structure of a group.
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| Several non-variety structures fail to be varieties, because either:
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| #It is necessary that 0 ≠ 1, 0 being the additive [[identity element]] and 1 being a multiplicative identity element, but this is a nonidentity;
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| #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.
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| Structures whose axioms unavoidably include nonidentities are among the most important ones in mathematics, e.g., [[Field (mathematics)|field]]s and hence also [[vector space]]s and [[algebra (ring theory)|algebra]]s. Although structures with nonidentities retain an undoubted algebraic flavor, they suffer from defects varieties do not have. For example, the product of two [[field (mathematics)|field]]s is not a field.
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| ==Category theory==
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| [[Category theory]] 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]], namely any [[function (mathematics)|function]] compatible with the operation(s) defining the structure. In this way, every algebraic structure gives rise to a [[category theory|category]]. For example, the [[category of groups]] has all [[Group (mathematics)|groups]] as objects and all [[group homomorphism]]s as morphisms. This [[concrete category]] may be seen as a [[category of sets]] with added category-theoretic [[structure (category theory)|structure]]. Likewise, the category of [[topological group]]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.
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| There are various concepts in category theory that try to capture the algebraic character of a context, for instance
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| *[[algebraic category]]
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| *[[essentially algebraic category]]
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| *[[presentable category]]
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| *[[locally presentable category]]
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| *[[Monad (category theory)|monadic]] functors and categories
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| *[[universal property]].
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| ==See also==
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| * [[Free object]]
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| * [[List of algebraic structures]]
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| * [[List of first order theories]]
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| * [[Signature (logic)|signature]]
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| * [[Structure (mathematical logic)]]
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| ==References==
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| {{reflist}}
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| * {{Citation | last1=Mac Lane | first1=Saunders | author1-link=Saunders Mac Lane | last2=Birkhoff | first2=Garrett | author2-link=Garrett Birkhoff | title=Algebra | publisher=AMS Chelsea | edition=2nd | isbn=978-0-8218-1646-2 | year=1999}}
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| * {{Citation | last1=Michel | first1=Anthony N. | last2=Herget | first2=Charles J. | title=Applied Algebra and Functional Analysis | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-67598-5 | year=1993}}
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| A monograph available online:
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| * {{Citation | last1=Burris | first1=Stanley N. | last2=Sankappanavar | first2=H. P. | title=A Course in Universal Algebra | url=http://www.thoralf.uwaterloo.ca/htdocs/ualg.html | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-90578-3 | year=1981}}
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| Category theory:
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| * {{Citation | last1=Mac Lane | first1=Saunders | author1-link=Saunders Mac Lane | title=[[Categories for the Working Mathematician]] | publisher=Springer-Verlag | location=Berlin, New York | edition=2nd | isbn=978-0-387-98403-2 | year=1998}}
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| * {{Citation | last1=Taylor | first1=Paul | title=Practical foundations of mathematics | publisher=[[Cambridge University Press]] | isbn=978-0-521-63107-5 | year=1999}}
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| ==External links==
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| * [http://math.chapman.edu/~jipsen/structures/doku.php Jipsen's algebra structures.] Includes many structures not mentioned here.
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| *[http://mathworld.wolfram.com/topics/Algebra.html Mathworld] page on abstract algebra.
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| *[[Stanford Encyclopedia of Philosophy]]: [http://plato.stanford.edu/entries/algebra/ Algebra] by [[Vaughan Pratt]].
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| [[Category:Abstract algebra]]
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| [[Category:Algebraic structures]]
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| [[Category:Mathematical structures]]
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