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| '''Idempotence''' ({{IPAc-en|ˌ|aɪ|d|ɨ|m|ˈ|p|oʊ|t|ən|s}} {{respell|EYE|dəm|POH|təns}}) is the property of certain operations in [[mathematics]] and [[computer science]], that can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of places in [[abstract algebra]] (in particular, in the theory of [[projector (linear algebra)|projector]]s and [[closure operator]]s) and [[functional programming]] (in which it is connected to the property of [[Referential transparency (computer science)|referential transparency]]).
| | Online marketing, also called advertising and marketing is forever evolving and merchandise in your articles own an online business you need to keep abreast of all alternatives available to you and enterprise. Building a business is a learning process and we all will get some things wrong. The main thing is to learn from our mistakes so as not to make them again. In my years of online marketing I have learned a lot and want to share with you, what I think, are 10 of the major online marketing mistakes.<br><br> |
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| The term was introduced by [[Benjamin Peirce]]<ref>Polcino & Sehgal (2002), p. 127.</ref> in the context of elements of algebras that remain invariant when raised to a positive integer power, and literally means "(the quality of having) the same power", from ''[[wikt:idem|idem]]'' + ''[[wikt:potence|potence]]'' (same + power).
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| There are several meanings of idempotence, depending on what the concept is applied to:
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| *A [[unary operation]] (or [[function (mathematics)|function]]) is idempotent if, whenever it is applied twice to any value, it gives the same result as if it were applied once; i.e., {{nowrap|''ƒ''(''ƒ''(''x'')) ≡ ''ƒ''(''x'')}}. For example, the [[absolute value]]: {{nowrap|abs(abs(''x'')) ≡ abs(''x'')}}.
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| *A [[binary operation]] is idempotent if, whenever it is applied to two equal values, it gives that value as the result. For example, the operation giving the [[maximum]] value of two values is idempotent: {{nowrap|max (''x'', ''x'') ≡ ''x''}}.
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| *Given a binary operation, an idempotent element (or simply an idempotent) for the operation is a value for which the operation, when given that value for both of its operands, gives the value as the result. For example, the number 1 is an idempotent of [[multiplication]]: {{nowrap|1 × 1 {{=}} 1}}.
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| == Definitions ==
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| ===Unary operation===
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| A [[unary operation]] <math>f</math>, that is, a map from some set <math>S</math> into itself, is called idempotent if, for all <math>x</math> in <math>S</math>,
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| :<math>f\!\left(f\!\left(x\right)\right) = f\!\left(x\right)</math>.
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| In particular, the [[identity function]] <math>\text{id}_S</math>, defined by <math>1 = \text{id}_S\left(x\right) = x</math>, is idempotent, as is the [[constant function]] <math>K_c</math>, where <math>c</math> is an element of <math>S</math>, defined by <math>1 = K_c\left(x\right) = c</math>.
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| An important class of idempotent functions is given by [[projection (linear algebra)|projection]]s in a [[vector space]]. An example of a projection is the function <math>\pi_{xy}</math> defined by <math>1 = \pi_{xy}\left(x, y, z\right) = \left(x, y, 0\right)</math>, which projects an arbitrary point in 3D space to a point on the <math>xy</math>-[[plane (mathematics)|plane]], where the third coordinate (<math>z</math>) is equal to 0.
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| A unary operation <math>f\colon S \to S</math> is idempotent if it maps each element of <math>S</math> to a [[Fixed point (mathematics)|fixed point]] of <math>f</math>. We can partition a set with <math>n</math> elements into <math>k</math> chosen fixed points and <math>n-k</math> non-fixed points, and then <math>k^{n-k}</math> is the number of different idempotent functions. Hence, taking into account all possible partitions,
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| :<math>\sum_{k=0}^n {n \choose k} k^{n-k}</math>
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| is the total number of possible idempotent functions on the set. The [[integer sequence]] of the number of idempotent functions as given by the sum above for <math>n = \left\{0, 1, 2, \dots\right\}</math> starts with <math>1, 1, 3, 10, 41, 196, 1057, 6322, 41393, \dots</math>. {{OEIS|A000248}}
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| ===Idempotent elements and binary operations===
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| {{main|Idempotent element}}
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| Given a [[binary operation]] <math>\bigstar</math> on a set <math>S</math>, an element <math>x</math> is said to be idempotent (with respect to <math>\bigstar</math>) if:
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| :<math>x \,\bigstar\, x = x</math>.
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| In particular an [[identity element]] of <math>\bigstar</math>, if it exists, is idempotent with respect to the operation <math>\bigstar</math>.
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| The binary operation itself is called idempotent if every element of <math>S</math> is idempotent. That is, for all <math>x \in S</math> when <math>\in</math> denotes set membership:
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| :<math>x \,\bigstar\, x = x</math>.
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| For example, the operations of [[union (set theory)|set union]] and [[intersection (set theory)|set intersection]] are both idempotent, as are [[logical conjunction]] and [[logical disjunction]], and, in general, the [[meet (mathematics)|meet]] and [[join (mathematics)|join]] operations of a [[lattice (order)|lattice]].
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| ===Connections===
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| The connections between the three notions are as follows.
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| *The statement that the binary operation ★ on a set ''S'' is idempotent, is equivalent to the statement that every element of ''S'' is idempotent for ★.
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| *The defining property of unary idempotence, {{nowrap|1=''f''(''f''(''x'')) = ''f''(''x'')}} for ''x'' in the domain of ''f'', can equivalently be rewritten as {{nowrap|1=''f'' ∘ ''f'' = ''f''}}, using the binary operation of [[function composition]] denoted by ∘. Thus, the statement that ''f'' is an idempotent unary operation on ''S'' is equivalent to the statement that ''f'' is an idempotent element with respect to the function composition operation ∘ on functions from ''S'' to ''S''.
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| ==Common examples==
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| === Functions ===
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| As mentioned above, the identity map and the constant maps are always idempotent maps. The [[absolute value]] function of a [[real number|real]] or [[complex number|complex]] argument, and the [[floor function]] of a real argument are idempotent.
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| The function that assigns to every subset <math>U</math> of some [[topological space]] <math>X</math>'' the [[closure (topology)|closure]] of <math>U</math> is idempotent on the [[power set]] <math>\mathcal{P}\left(X\right)</math> of <math>X</math>. It is an example of a [[closure operator]]; all closure operators are idempotent functions.
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| The operation of subtracting the average of a list of numbers from every number in the list is idempotent. For example, consider the numbers <math>3, 6, 8, 8, \text{and }10</math>. The average <math>\frac{\sum_1^n x_n}{n} \,\forall x_n</math> is <math>\frac{3 + 6 + 8 + 8 + 10}{5} = \frac{35}{5} = 7</math>. Subtracting 7 from every number in the list yields <math>\left(-4\right), \left(-1\right), 1, 1, 3</math>. The average <math>\frac{\sum_1^n x_n}{n} \,\forall x_n</math> of that list is <math>\frac{\left(-4\right) + \left(-1\right) + 1 + 1 +3}{5} = \frac{0}{5} = 0</math>. Subtracting 0 from every number in that list yields the same list.
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| === Formal languages ===
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| The [[Kleene star]] and [[Kleene plus]] operators used to express repetition in [[Formal grammar|formal languages]] are idempotent.
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| === Idempotent ring elements ===
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| {{main |Idempotent element}}
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| An idempotent element of a [[ring (mathematics)|ring]] is, by definition, an element that is idempotent for the ring's multiplication.<ref>See [[Michiel Hazewinkel|Hazewinkel]] et al. (2004), p. 2.</ref> That is, for an idempotent element <math>a</math>, <math>a^2 = a</math>.
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| Idempotent elements of rings yield [[Indecomposable_module|direct decomposition]]s of [[module (algebra)|modules]], and play a role in describing other homological properties of the ring.
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| While "idempotent" usually refers to the multiplication operation of a ring, there are rings in which both operations are idempotent: [[Boolean algebra]]s are such an example.
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| === Other examples ===
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| In [[Boolean algebra]], both the [[Logical conjunction|logical and]] and the [[Logical disjunction|logical or]] operations are idempotent. This implies that every element of Boolean algebra is idempotent with respect to both of these operations. Specifically, <math>x \wedge x = x</math> and <math>x \vee x = x \forall x</math> where <math>\forall</math> stands for 'for all.'
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| In [[linear algebra]], [[projection (linear algebra)|projection]]s are idempotent. In fact, the projections of a vector space are exactly the idempotent elements of the ring of [[linear transformation]]s of the vector space. After fixing a [[basis (linear algebra)|basis]], it can be shown that the matrix of a projection with respect to this basis is an idempotent matrix.
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| An [[idempotent semiring]] (also sometimes called a ''dioid'') is a semiring whose ''addition'' (not multiplication) is idempotent. If both operations of the semiring are idempotent, then the semiring is called ''doubly idempotent''.<ref>Gondran & Minoux. ''Graphs, dioids and semirings''. Springer, 2008, p. 34</ref>
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| ==Computer science meaning==
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| {{See also|Referential transparency (computer science)|Reentrant (subroutine)|Stable sort}}
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| In [[computer science]], the term '''idempotent''' is used more comprehensively to describe an operation that will produce the same results if executed once or multiple times.<ref name=IBM>{{cite web|last=Rodriguez|first=Alex|title=RESTful Web services: The basics|url=https://www.ibm.com/developerworks/webservices/library/ws-restful/|work=IBM developerWorks|publisher=IBM|accessdate=24 April 2013}}</ref> This may have a different meaning depending on the context in which it is applied. In the case of [[Method (computer science)|method]]s or [[subroutine]] calls with [[Side effect (computer science)|side effects]], for instance, it means that the modified state remains the same after the first call. In [[functional programming]], though, an idempotent function is one that has the property {{nowrap|1=''f''(''f''(''x'')) = ''f''(''x'')}} for any value ''x''.<ref>http://foldoc.org/idempotent</ref>
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| This is a very useful property in many situations, as it means that an operation can be repeated or retried as often as necessary without causing unintended effects. With non-idempotent operations, the algorithm may have to keep track of whether the operation was already performed or not.
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| === Examples ===
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| Looking up some customer's name and address in a [[database]] are typically idempotent (in fact [[wiktionary:nullipotent|''nullipotent'']]), since this will not cause the database to change. Similarly, changing a customer's address is typically idempotent, because the final address will be the same no matter how many times it is submitted. However, placing an order for a car for the customer is typically not idempotent, since running the method/call several times will lead to several orders being placed. Canceling an order is idempotent, because the order remains canceled no matter how many requests are made.
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| A composition of idempotent methods or subroutines, however, is not necessarily idempotent if a later method in the sequence changes a value that an earlier method depends on – idempotence is not closed under composition. For example, suppose the initial value of a variable is 3 and there is a sequence that reads the variable, then changes it to 5, and then reads it again. Each step in the sequence is idempotent: both steps reading the variable have no side effects and changing a variable to 5 will always have the same effect no matter how many times it is executed. Nonetheless, executing the entire sequence once produces the output (3, 5), but executing it a second time produces the output (5, 5), so the sequence is not idempotent.<ref name="httpStd-methods">W3C, [http://www.w3.org/Protocols/rfc2616/rfc2616-sec9.html HyperText Transfer Protocol v. 1.1 Methods]. See also [[Hypertext Transfer Protocol|HyperText Transfer Protocol]].</ref>
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| In the [[Hypertext Transfer Protocol|HyperText Transfer Protocol (HTTP)]], idempotence and [[Hypertext Transfer Protocol#Safe methods|safety]] are the major attributes that separate HTTP verbs. Of the major HTTP verbs, GET, PUT, and DELETE are idempotent (if implemented according to the standard), but POST is not.<ref name="httpStd-methods" /> These verbs represent very abstract operations in computer science: GET retrieves a resource; PUT stores content at a resource; and DELETE eliminates a resource. As in the example above, reading data usually has no side effects, so it is idempotent (in fact nullipotent). Storing a given set of content is usually idempotent, as the final value stored remains the same after each execution. And deleting something is generally idempotent, as the end result is always the absence of the thing deleted.
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| In [[Event Stream Processing]], idempotence refers to the ability of a system to produce the same outcome, even if an event or message is received more than once.
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| In a [[load-store architecture]], instructions that might possibly cause a [[page fault]] are idempotent. So if a page fault occurs, the OS can load the page from disk and then simply re-execute the faulted instruction.
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| In a processor where such instructions are not idempotent, dealing with page faults is [[Motorola 68000#Interrupts|much more complex]].
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| ==Applied examples==
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| Applied examples that many people could encounter in their day-to-day lives include [[elevator]] call buttons and crosswalk buttons.<ref>http://web.archive.org/web/20110523081716/http://www.nclabor.com/elevator/geartrac.pdf For example, this design specification includes detailed algorithm for when elevator cars will respond to subsequent calls for service</ref> The initial activation of the button moves the system into a requesting state, until the request is satisfied. Subsequent activations of the button between the initial activation and the request being satisfied have no effect. Many people will subsequently activate an idempotent button, even if consciously aware of the rational information that it will have no effect.
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| ==See also==
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| *[[Closure operator]]
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| *[[Fixed point (mathematics)]]
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| *[[Idempotent of a code]]
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| *[[Nilpotent]]
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| *[[Idempotent matrix]]
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| *[[List of matrices]]
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| *[[Pure function]]
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| *[[Referential transparency (computer science)]]
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| *[[Iterated function]]
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| *[[Biordered set]]
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| *[[Involution (mathematics)]]
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| ==References==
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| {{Reflist}}
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| == Further reading ==
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| * “[http://foldoc.org/idempotent idempotent]” at [[FOLDOC]]
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| *{{citation
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| |author=Goodearl, K. R.
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| |title=von Neumann regular rings
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| |edition=2
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| |publisher=Robert E. Krieger Publishing Co. Inc.
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| |place=Malabar, FL
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| |year=1991
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| |pages=xviii+412
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| |isbn=0-89464-632-X
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| |mr=1150975 (93m:16006)}}
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| *{{citation
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| |author1=[[Hazewinkel, Michiel]]
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| |author2=Gubareni, Nadiya
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| |author3=Kirichenko, V. V.
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| |title=Algebras, rings and modules. vol. 1
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| |series=Mathematics and its Applications
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| |volume=575
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| |publisher=Kluwer Academic Publishers
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| |place=Dordrecht
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| |year=2004
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| |pages=xii+380
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| |isbn=1-4020-2690-0
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| |mr=2106764 (2006a:16001)}}
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| *{{citation
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| |author=Lam, T. Y.
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| |title=A first course in noncommutative rings
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| |series=Graduate Texts in Mathematics
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| |volume=131
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| |edition=2
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| |publisher=Springer-Verlag
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| |place=New York
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| |year=2001
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| |pages=xx+385
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| |isbn=0-387-95183-0
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| |mr=1838439 (2002c:16001)}}
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| * {{Lang Algebra|edition=3}} p. 443
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| * Peirce, Benjamin. [http://www.math.harvard.edu/history/peirce_algebra/index.html ''Linear Associative Algebra''] 1870.
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| *{{citation
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| |author1=Polcino Milies, César
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| |author2=Sehgal, Sudarshan K.
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| |title=An introduction to group rings
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| |series=Algebras and Applications
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| |volume=1
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| |publisher=Kluwer Academic Publishers
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| |place=Dordrecht
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| |year=2002
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| |pages=xii+371
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| |isbn=1-4020-0238-6
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| |mr=1896125 (2003b:16026)}}
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| == External links ==
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| {{wiktionary}}
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| * {{springer|title=Idempotent|id=p/i050080}}
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| {{wikibooks}}
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| {{wikiversity | Portal:Computer Science}}
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| [[Category:Abstract algebra]]
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| [[Category:Closure operators]]
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| [[Category:Mathematical relations]]
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| [[Category:Theoretical computer science]]
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| [[Category:Binary operations]]
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