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| In [[abstract algebra]], a '''semiring''' is an [[algebraic structure]] similar to a [[Ring (algebra)|ring]], but without the requirement that each element must have an [[additive inverse]]. The term '''rig''' is also used occasionally<ref>Głazek (2002) p.7</ref> — this originated as a joke, suggesting that rigs are ri''n''gs without ''n''egative elements.
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| {{Algebraic structures |Ring}}
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| == Definition ==
| | Looking at playing a new visual game, read the take advantage of book. Most betting games have a book you'll can purchase separately. You may want to help you consider doing this as well as , reading it before your corporation play, or even despite the fact you are playing. This way, you can get the most out of your game play.<br><br>Interweaving social styles form that strong net in which we are all caught up. When The Tygers of Pan Tang sang 'It's lonely at the top. Everybody's undertaking to do you in', these people borrowed drastically from clash of clans hack tool no [http://Answers.yahoo.com/search/search_result?p=survey+form&submit-go=Search+Y!+Answers survey form]. A society while not clash of clans hack tool no survey are like a society and no knowledge, in which often it is quite awesome.<br><br>Be aware of how multi player works. Should you're investing in a real game exclusively for unique multiplayer, be sure individuals have everything required to suit this. If you're the one planning on playing in the direction of a person in your household, you may stumble on that you will are after two copies of the very clash of clans cheats to master against one another.<br><br>Guilds and clans have ended up popular ever since the very beginning of first-person gift idea shooter and MMORPG avid gamers. World of WarCraft develops for that concept with their personally own World associated Warcraft guilds. A real guild can easily always prove understood as a in respect of players that band away for companionship. People in the guild travel back together again for fun and excite while improving in trial and gold.<br><br>Once the game is a fabulous mobile edition, it may not lack substance comparable to many mobile games. So, defragging the process registry will boost system overall performance so that you a fantastic extent. I usually get anywhere you want to from 4000 to 6000 m - Points in the day ($4 to $5 for Amazon. Apple mackintosh showed off the vastly anticipated i - Some of the 5 for the incredibly first time in San Francisco on Wednesday morning (September 12, 2012). Can be certainly a huge demand for some i - Mobile device 4 application not alone promoting business but potentially helps users to secure extra money.<br><br>If you are the proud owner of an ANY easily transportable device that runs on iOS or android basically a touchscreen tablet equipment or a smart phone, then you definitely definitely have already been very careful of the revolution taking place right now the actual world world of mobile computer game "The Clash Regarding Clans", and you would expect to be in demand pointing to conflict of families free jewels compromise because good deal more gems, elixir and valuable metal are needed seriously toward acquire every battle.<br><br>Future house fires . try interpreting the particular abstracts differently. Hope of it in offer of bulk with other jewels to skip 1 moment. Skipping added your time expenses added money, but rather you get a grander deal. Think to do with it as a only a handful of accretion discounts In case you adored this article and you would want to be given more details concerning [http://circuspartypanama.com hack clash of clans iphone] generously pay a visit to our web site. . |
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| A '''semiring''' is a [[Set (mathematics)|set]] ''R'' equipped with two [[binary operation]]s + and ·, called addition and multiplication, such that:<ref>Berstel & Perrin (1985), {{Google books quote|id=GHJHqezwwpcC|page=26|text=a semiring K is a set equipped with two operations|p. 26}}</ref><ref name=LotIII211>Lothaire (2005) p.211</ref><ref name=Sak2728>Sakarovitch (2009) pp.27–28</ref>
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| # (''R'', +) is a [[commutative monoid]] with [[identity element]] 0:
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| ## (''a'' + ''b'') + ''c'' = ''a'' + (''b'' + ''c'')
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| ## 0 + ''a'' = ''a'' + 0 = ''a''
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| ## ''a'' + ''b'' = ''b'' + ''a''
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| # (''R'', ·) is a [[monoid]] with identity element 1:
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| ## (''a''·''b'')·''c'' = ''a''·(''b''·''c'')
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| ## 1·''a'' = ''a''·1 = ''a''
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| # Multiplication left and right [[distributive law|distributes]] over addition:
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| ## ''a''·(''b'' + ''c'') = (''a''·''b'') + (''a''·''c'')
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| ## (''a'' + ''b'')·''c'' = (''a''·''c'') + (''b''·''c'')
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| # Multiplication by 0 annihilates ''R'':
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| ## 0·''a'' = ''a''·0 = 0
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| This last [[axiom]] is omitted from the definition of a [[ring (algebra)|ring]]: it follows automatically from the other ring axioms. Here it does not, and it is necessary to state it in the definition.
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| The difference between rings and semirings, then, is that addition yields only a [[commutative monoid]], not necessarily a [[commutative group]]. Specifically, elements in semirings do not necessarily have an inverse for the addition.
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| The symbol · is usually omitted from the notation; that is, ''a''·''b'' is just written ''ab''. Similarly, an [[order of operations]] is accepted, according to which · is applied before +; that is, {{nowrap|''a'' + ''bc''}} is {{nowrap|''a'' + (''bc'').}}
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| A '''commutative semiring''' is one whose multiplication is [[commutative]].<ref name=LotIII212>Lothaire (2005) p.212</ref> An '''idempotent semiring''' (also known as a '''dioid''') is one whose ''addition'' is [[idempotent]]: ''a'' + ''a'' = ''a'', that is, (''R'', +, 0) is a [[semilattice#Semilattices as algebraic structures|join-semilattice with zero]].
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| There are some authors who prefer to leave out the requirement that a semiring have a 0 or 1. This makes the analogy between ''[[ring (algebra)|ring]]'' and ''semiring'' on the one hand and ''[[group (mathematics)|group]]'' and ''[[semigroup]]'' on the other hand work more smoothly. These authors often use ''rig'' for the concept defined here.<ref group="note">[http://www.proofwiki.org/wiki/Definition:Rig For an example see the definition of rig on Proofwiki.org]</ref>
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| == Examples == | |
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| ===In general===
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| * Any ring is also a semiring.
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| * The [[ideal (ring theory)|ideals]] of a ring form a semiring under addition and multiplication of ideals.
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| * Any [[quantale|unital quantale]] is an idempotent semiring, or dioid, under join and multiplication.
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| * Any bounded, [[distributive lattice]] is a commutative, idempotent semiring under join and meet.
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| * In particular, a [[Boolean algebra (structure)|Boolean algebra]] is such a semiring. A [[Boolean ring]] is also a semiring—indeed, a ring—but it is not idempotent under ''addition''. A ''Boolean semiring'' is a semiring isomorphic to a subsemiring of a Boolean algebra.<ref>{{cite book | title=Surveys in Contemporary Mathematics | volume=347 | series=London Mathematical Society Lecture Note Series | issn=0076-0552 | editor1-first=Nicholas | editor1-last=Young | editor2-first=Yemon | editor2-last=Choi | publisher=[[Cambridge University Press]] | year=2008 | isbn=0-521-70564-9 | chapter=Rank and determinant functions for matrices over semirings | first=Alexander E. | last=Guterman | pages=1-33 | zbl=1181.16042 }}</ref>
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| * A normal [[skew lattice]] in a ring ''R'' is an idempotent semiring for the operations multiplication and nabla, where the latter operation is defined by <math>a\nabla b=a+b+ba-aba-bab</math>.
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| * Any [[c-semiring]] is also a semiring, where addition is idempotent and defined over arbitrary sets.
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| ===Specific examples===
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| * The motivating example of a semiring is the set of [[natural number]]s '''N''' (including [[0 (number)|zero]]) under ordinary addition and multiplication. Likewise, the non-negative [[rational number]]s and the non-negative [[real number]]s form semirings. All these semirings are commutative.
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| * The square ''n''-by-''n'' [[matrix (mathematics)|matrices]] with non-negative entries form a (non-commutative) semiring under ordinary addition and multiplication of matrices. More generally, this likewise applies to the square matrices whose entries are elements of any other given semiring ''S'', and the semiring is generally non-commutative even though ''S'' may be commutative.
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| * If ''A'' is a commutative monoid, the set ''End(A)'' of [[endomorphism]]s ''f:A→A'' form a semiring, where addition is pointwise addition and multiplication is [[function composition]]. The [[zero morphism]] and the identity are the respective neutral elements. If ''A'' is the additive monoid of natural numbers we obtain the semiring of natural numbers as ''End(A)'', and if ''A=S^n'' with ''S'' a semiring, we obtain (after associating each morphism to a matrix) the semiring of square ''n''-by-''n'' matrices with coefficients in ''S''.
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| * The commutative semiring '''B''' formed by the [[two-element Boolean algebra]]:<ref name=LotIII211/> this is the simplest example of a semiring which is not a ring.
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| * '''N'''[x], [[polynomial]]s with natural number coefficients form a commutative semiring. In fact, this is the [[free object|free]] commutative semiring on a single generator {''x''}.
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| * Of course, rings such as the [[integer]]s or the [[real number]]s are also examples of semirings.
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| * The [[tropical semiring]], '''R''' ∪ {−∞}, is a commutative, idempotent semiring with max(''a'',''b'') serving as semiring addition (identity −∞) and ordinary addition (identity 0) serving as semiring multiplication. In an alternative formulation, the tropical semiring is '''R''' ∪ {∞}, and min replaces max as the addition operation.<ref name=LotIII211/>
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| * The set of [[cardinal number]]s smaller than any given [[Infinity|infinite]] cardinal form a semiring under cardinal addition and multiplication. The set of ''all cardinals'' of an [[inner model]] form a semiring under (inner model) cardinal addition and multiplication.
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| * The '''probability semiring''' of non-negative real numbers under the usual addition and multiplication.<ref name=LotIII211/>
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| * The '''log semiring''' on '''R''' ∪ ±∞ with addition given by
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| :<math> x \oplus y = - \log(e^{-x}+e^{-y}) \ , </math>
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| :with multiplication +, zero element +∞ and unit element 0.<ref name=LotIII211/>
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| * The family of (isomorphism equivalence classes of) [[combinatorial class]]es (sets of countably many objects with non-negative integer sizes such that there are finitely many objects of each size) with the empty class as the zero object, the class consisting only of the empty set as the unit, [[disjoint union]] of classes as addition, and [[Cartesian product]] of classes as multiplication.<ref>{{citation|title=Algebraic Cryptanalysis|first=Gregory V.|last=Bard|publisher=Springer|year=2009|isbn=9780387887579|at=Section 4.2.1, "Combinatorial Classes", ff., pp. 30–34|url=http://books.google.com/books?id=kjbp0mgu3IAC&pg=PA30}}.</ref>
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| == Semiring theory ==
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| Much of the theory of rings continues to make sense when applied to arbitrary semirings.
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| In particular, one can generalise the theory of [[algebra (ring theory)|algebras]] over [[commutative ring]]s directly to a theory of algebras over commutative semirings.
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| Then a ring is simply an algebra over the commutative semiring '''Z''' of [[integer]]s.
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| Some mathematicians go so far as to say that semirings are really the more fundamental concept, and specialising to rings should be seen in the same light as specialising to, say, algebras over the [[complex number]]s.
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| Idempotent semirings are special to semiring theory as any ring which is idempotent under addition is trivial. One can define a [[partial order]] ≤ on an idempotent semiring by setting ''a'' ≤ ''b'' whenever ''a'' + ''b'' = ''b'' (or, equivalently, if there exists an ''x'' such that ''a'' + ''x'' = ''b''). It is easy to see that 0 is the [[least element]] with respect to this order: 0 ≤ ''a'' for all ''a''. Addition and multiplication respect the ordering in the sense that ''a'' ≤ ''b'' implies ''ac'' ≤ ''bc'' and ''ca'' ≤ ''cb'' and (''a''+''c'') ≤ (''b''+''c'').
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| ==Applications==
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| Dioids, especially the (max, +) and (min, +) dioids on the reals, are often used in [[performance evaluation]] on discrete event systems. The real numbers then are the "costs" or "arrival time"; the "max" operation corresponds to having to wait for all prerequisites of an events (thus taking the maximal time) while the "min" operation corresponds to being able to choose the best, less costly choice; and + corresponds to accumulation along the same path.
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| The [[Floyd–Warshall algorithm]] for [[shortest path]]s can thus be reformulated as a computation over a (min, +) algebra. Similarly, the [[Viterbi algorithm]] for finding the most probable state sequence corresponding to an observation sequence in a [[Hidden Markov model]] can also be formulated as a computation over a (max, ×) algebra on probabilities. These [[dynamic programming]] algorithms rely on the [[distributive property]] of their associated semirings to compute quantities over a large (possibly exponential) number of terms more efficiently than enumerating each of them.
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| ==Starsemirings==
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| A '''starsemiring''' is a semiring with an additional unary operator * ([[Kleene star]]).<ref name=BR27>Berstel & Reutenauer (2011) p.27</ref> A '''[[Kleene algebra]]''' is a starsemiring with idempotent addition: they are important in the theory of [[formal language]]s and [[regular expression]]s. A '''Conway semiring''' is a starsemiring satisfying the sum-star and the product-star equations:<ref>{{cite book | last1=Ésik | first1=Zoltán | last2=Kuich | first2=Werner | chapter=Equational axioms for a theory of automata | editor1-last=Martín-Vide | editor1-first=Carlos | title=Formal languages and applications | location=Berlin | publisher=[[Springer-Verlag]] | series=Studies in Fuzziness and Soft Computing | volume=148 | pages=183–196 | year=2004 | isbn=3-540-20907-7 | zbl=1088.68117 }}</ref>
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| :<math>(a+b)^* = (a^*b)^*a^*,\,</math>
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| :<math>(ab)^* = 1 + a(ba)^*b.\,</math>
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| == Further generalizations ==
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| A ''[[near-semiring|near-ring]]'' does not require addition to be commutative, nor does it require right-distributivity. Just as cardinal numbers form a semiring, so do [[ordinal number]]s form a [[near-ring]].
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| In [[category theory]], a ''[[2-rig]]'' is a category with [[functor]]ial operations analogous to those of a rig. That the cardinal numbers form a rig can be categorified to say that the [[category of sets]] (or more generally, any [[topos]]) is a 2-rig. | |
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| ==Semiring of sets==
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| A '''semiring (of sets)'''<ref>Noel Vaillant, [http://www.probability.net/WEBcaratheodory.pdf Caratheodory's Extension], on probability.net.</ref> is a non-empty collection S of sets such that
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| # <math>\emptyset \in S</math>
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| # If <math>E \in S</math> and <math>F \in S</math> then <math>E \cap F \in S</math>.
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| # If <math>E \in S</math> and <math>F \in S</math> then there exists a finite number of mutually [[disjoint sets]] <math>C_i \in S</math> for <math>i=1,\ldots,n</math> such that <math>E \setminus F = \bigcup_{i=1}^n C_i</math>.
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| Such semirings are used in measure theory. An example of a semiring of sets is the collection of half-open, half-closed real [[Interval (mathematics)|intervals]] <math>[a,b) \subset \mathbb{R}</math>.
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| ==See also==
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| *[[Ring (algebra)]]
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| *[[Ring of sets]]
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| ==Notes==
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| {{reflist|group=note}}
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| ==Bibliography==
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| <references />
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| * [[François Baccelli]], Guy Cohen, Geert Jan Olsder, Jean-Pierre Quadrat, ''[http://cermics.enpc.fr/~cohen-g//SED/book-online.html Synchronization and Linearity (online version)]'', Wiley, 1992, ISBN 0-471-93609-X
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| * Golan, Jonathan S., ''Semirings and their applications''. Updated and expanded version of ''The theory of semirings, with applications to mathematics and theoretical computer science'' (Longman Sci. Tech., Harlow, 1992, {{MathSciNet|id=1163371}}. Kluwer Academic Publishers, Dordrecht, 1999. xii+381 pp. ISBN 0-7923-5786-8 {{MathSciNet|id=1746739}}
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| * {{cite book |last1= Berstel |first1=Jean |authorlink1= |last2=Perrin |first2=Dominique |authorlink2= |title=Theory of codes |url= |edition= |series=Pure and applied mathematics |volume=117 |year=1985 |publisher=Academic Press |location= |isbn=978-0-12-093420-1 | zbl=0587.68066 }}
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| *{{cite book | last=Lothaire | first=M. | authorlink=M. Lothaire | title=Applied combinatorics on words | others=A collective work by Jean Berstel, Dominique Perrin, Maxime Crochemore, Eric Laporte, Mehryar Mohri, Nadia Pisanti, Marie-France Sagot, Gesine Reinert, Sophie Schbath, Michael Waterman, Philippe Jacquet, Wojciech Szpankowski, Dominique Poulalhon, Gilles Schaeffer, Roman Kolpakov, Gregory Koucherov, Jean-Paul Allouche and Valérie Berthé| series=Encyclopedia of Mathematics and Its Applications | volume=105 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2005 | isbn=0-521-84802-4 | zbl=1133.68067 }}
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| *{{cite book | last=Głazek | first=Kazimierz | title=A guide to the literature on semirings and their applications in mathematics and information sciences. With complete bibliography | location=Dordrecht | publisher=Kluwer Academic | year=2002 | isbn=1-4020-0717-5 | zbl=1072.16040 }}
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| * {{cite book | last=Sakarovitch | first=Jacques | title=Elements of automata theory | others=Translated from the French by Reuben Thomas | location=Cambridge | publisher=[[Cambridge University Press]] | year=2009 | isbn=978-0-521-84425-3 | zbl=1188.68177 }}
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| * {{cite book | last1=Berstel | first1=Jean | last2=Reutenauer | first2=Christophe | title=Noncommutative rational series with applications | series=Encyclopedia of Mathematics and Its Applications | volume=137 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2011 | isbn=978-0-521-19022-0 | zbl=1250.68007 }}
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| [[Category:Algebraic structures]]
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| [[Category:Ring theory]]
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