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| In [[abstract algebra]], the '''free monoid''' on a [[Set (mathematics)|set]] ''A'' is the [[monoid]] whose elements are all the finite sequences (or [[character string|strings]]) of zero or more elements from ''A''. It is usually denoted ''A''<sup>∗</sup>. The [[identity element]] is the unique sequence of zero elements, often called the [[empty string]] and denoted by ε or λ, and the monoid operation is [[string concatenation]]. The '''free [[semigroup]]''' on ''A'' is the subsemigroup of ''A''<sup>∗</sup> containing all elements except the empty string. It is usually denoted ''A''<sup>+</sup>.<ref name=Lot23>{{harvtxt|Lothaire|1997|pp=2–3}}, [http://books.google.com/books?id=eATLTZzwW-sC&pg=PA2]</ref><ref name=PF2>{{harvtxt|Pytheas Fogg|2002|p=2}}</ref>
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| More generally, an abstract monoid (or semigroup) ''S'' is described as '''free''' if it is [[isomorphic]] to the free monoid (or semigroup) on some set.<ref name=Lot5>{{harvtxt|Lothaire|1997|p=5}}</ref>
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| As the name implies, free monoids and semigroups are those objects which satisfy the usual [[universal property]] defining [[free object]]s, in the respective [[category (mathematics)|categories]] of monoids and semigroups. It follows that every monoid (or semigroup) arises as a homomorphic image of a free monoid (or semigroup). The study of semigroups as images of free semigroups is called combinatorial semigroup theory.
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| == Examples ==
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| === Natural numbers ===
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| The monoid ('''N''',+) of [[natural numbers]] (including zero) under addition is a free monoid on a singleton free generator, in this case the natural number 1.
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| According to the formal definition, this monoid consists of all sequences like "1", "1+1", "1+1+1", "1+1+1+1", and so on, including the empty sequence.
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| Mapping each such sequence to its evaluation result
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| <ref>Since addition of natural numbers is associative, the result doesn't depend on the order of evaluation, thus ensuring the mapping to be well-defined.</ref>
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| and the empty sequence to zero establishes an isomorphism from the set of such sequences to '''N'''. | |
| This isomorphism is compatible with "+", that is, for any two sequences ''s'' and ''t'', if ''s'' is mapped (i.e. evaluated) to a number ''m'' and ''t'' to ''n'', then their concatenation ''s''+''t'' is mapped to the sum ''m''+''n''.
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| === Kleene star ===
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| In [[formal language]] theory, usually a finite set (called "alphabet" there) ''A'' of "symbols" is considered, a finite sequence of symbols is called "word over ''A''", and the free monoid ''A''<sup>∗</sup> is called the "[[Kleene star]] of ''A''".
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| Thus, the abstract study of formal languages can be thought of as the study of subsets of finitely generated free monoids. There are deep connections between the theory of semigroups and that of [[automata theory|automata]]. For example, the [[regular language]]s over ''A'' are the homomorphic pre-images in ''A''<sup>∗</sup> of subsets of finite monoids.{{clarify|reason=At least, a reference to a definition of 'homomorphic pre-image' and a reference to a proof should be added here. Also, rephrasing the sentence in singular may improve clarity.|date=August 2013}}
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| For example, assuming an alphabet ''A'' = {''a'', ''b'', ''c''}, its Kleene star ''A''<sup>∗</sup> contains all concatenations of ''a'', ''b'', and ''c'':
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| :{ε, ''a'', ''ab'', ''ba'', ''caa'', ''cccbabbc'', ...}.
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| If ''A'' is any set, the ''word length'' function on ''A''<sup>∗</sup> is the unique [[monoid homomorphism]] from ''A''<sup>∗</sup> to ('''N''',+) that maps each element of ''A'' to 1. A free monoid is thus a '''graded monoid'''.{{clarify|reason=Give a definition of 'graded monoid' or a reference to it. A Wikipedia article 'graded ring' exists, but it doesn't define graded monoids. In case 'graded' is meant just informally, the sentence should better be rephrased as e.g. 'Due to this additional functionality, some authors call a free monoid a ⟨graded⟩ one.'|date=August 2013}}<ref name=Sak382>Sakarovitch (2009) p.382</ref>
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| ==Conjugate words==
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| [[File:Example of strings equidivisibility.gif|thumb|Example for 1st case of equidivisibility: m="UNCLE", n="ANLY", p="UN", q="CLEANLY", and s="CLE"]]
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| We define a pair of words in ''A''<sup>∗</sup> of the form ''uv'' and ''vu'' as '''conjugate''': the conjugates of a word are thus its [[circular shift]]s.<ref name=Sak27>Sakarovitch (2009) p.27</ref> Two words are conjugate in this sense if they are [[Conjugation (group theory)|conjugate in the sense of group theory]] as elements of the [[free group]] generated by ''A''.<ref name=PF297>{{harvtxt|Pytheas Fogg|2002|p=297}}</ref>
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| ===Equidivisibility===
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| A free monoid is '''equidivisible''': if the equation ''mn'' = ''pq'' holds, then there exists an ''s'' such that either ''m'' = ''ps'', ''sn'' = ''q'' (example see image) or ''ms'' = ''p'', ''n'' = ''sq''.<ref name=Sak26>Sakarovitch (2009) p.26</ref>
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| A monoid is free if and only if it is graded and equidivisible.<ref name=Sak26/>
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| == Free generators and rank ==
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| The members of a set ''A'' are called the '''free generators''' for ''A''<sup>∗</sup> and ''A''<sup>+</sup>. The superscript * is then commonly understood to be the [[Kleene star]]. More generally, if ''S'' is an abstract free monoid (semigroup), then a set of elements which maps onto the set of single-letter words under an isomorphism to a semigroup ''A<sup>+</sup>'' (monoid ''A''<sup>∗</sup>) is called a ''set of free generators'' for ''S''.
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| Each free semigroup (or monoid) ''S'' has exactly one set of free generators, the [[cardinality]] of which is called the ''rank'' of ''S''.
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| Two free monoids or semigroups are isomorphic if and only if they have the same rank. In fact, ''every'' set of generators for a free semigroup or monoid ''S'' contains the free generators.{{clarify|reason=What is the difference between a set of generators and a set of free generators? Up to here, only the latter has been defined.|date=August 2013}} It follows that a free semigroup or monoid is finitely generated if and only if it has finite rank.
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| A [[Submonoid#Generators_and_submonoids|submonoid]] ''N'' of ''A''<sup>∗</sup> is '''stable''' if ''u'', ''v'', ''ux'', ''xv'' in ''N'' together imply ''x'' in ''N''.<ref name=BPR61>{{harvtxt|Berstel|Perrin|Reutenauer|2010|p=61}}</ref> A submonoid of ''A''<sup>∗</sup> is stable if and only if it is free.<ref name=BPR62>{{harvtxt|Berstel|Perrin|Reutenauer|2010|p=62}}</ref>
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| For example, using the set of [[bit]]s { "0", "1" } as ''A'', the set ''N'' of all bit strings containing evenly many "1"s is a stable<ref>if ''u'' contains an even number of "1"s, and ''ux'' as well, then ''x'' must contain an even number of "1"s, too</ref> submonoid<ref>since it is closed with respect to string concatenation</ref> of the set ''A''<sup>∗</sup> of all bit strings at all. While ''N'' cannot be freely generated by any set of single bits, it ''can'' be freely generated by the set of bit strings { "0", "11", "101", "1001", "10001", ... }.{{clarify|reason=Somebody please cross-check my example: Aren't generator sets restricted to single-character sets, so my generator set wouldn't be valid? But I can't imagine another one. On the other hand, each member of N can obviously be decomposed into a sequence of my generators in a unique way, e.g. 1010101 → 101⋅0⋅101|date=August 2013}}
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| ===Codes===
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| A set of free generators for a free monoid ''P'' is referred to as a '''basis''' for '''P''': a set of words ''C'' is a '''code''' if ''C''* is a free monoid and ''C'' is a basis.<ref name=Lot5/> A set ''X'' of words in ''A''<sup>∗</sup> is a '''prefix''', or has the '''prefix property''', if it does not contain a proper [[prefix (computer science)|(string) prefix]] of any of its elements<!--- deleted, since "≤" as shorthand for "string prefix of" has not been defined yet, and is not used elsewhere in the article: that is, ''x'',''y'' in ''X'' with ''x'' ≤ ''y'' implies ''x'' = ''y''--->. Every prefix in ''A''<sup>+</sup> is a code, indeed a [[prefix code]].<ref name=Lot5/><ref name=BPR58>{{harvtxt|Berstel|Perrin|Reutenauer|2010|p=58}}</ref>
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| A submonoid ''N'' of ''A''<sup>∗</sup> is '''right unitary''' if ''x'', ''xy'' in ''N'' implies ''y'' in ''N''. A submonoid is generated by a prefix if and only if it is right unitary.<ref name=Lot15>{{harvtxt|Lothaire|1997|p=15}}</ref>
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| ==Free hull==
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| The intersection of free submonoids of a free monoid ''A''<sup>∗</sup> is again free.<ref name=Lot6>{{harvtxt|Lothaire|1997|p=6}}</ref><ref name=LotII204>{{harvtxt|Lothaire|2011|p=204}}</ref> If ''S'' is a subset of a free monoid ''A''* then the intersection of all free submonoids of ''A''* containing ''S'' is well-defined, since ''A''* itself is free, and contains ''S''; it is a free monoid. A basis for this intersection is the '''free hull''' of ''S''.
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| The '''defect theorem'''<ref name=Lot6/><ref name=LotII204/><ref name=BPR66>{{harvtxt|Berstel|Perrin|Reutenauer|2010|p=66}}</ref> states that if ''X'' is finite and ''C'' is the free hull of ''X'', then either ''X'' is a code and ''C'' = ''X'', or
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| :|''C''| ≤ |''X''| − 1 .
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| ==Morphisms==
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| A [[monoid morphism]] ''f'' from a free monoid ''B''<sup>∗</sup> to a monoid ''M'' is a map such that ''f''(''xy'') = ''f''(''x'')⋅''f''(''y'') for words ''x'',''y'' and ''f''(ε) = ι, where ε and ι denotes the identity element of ''B''<sup>∗</sup> and ''M'', respectively. The morphism ''f'' is determined by its values on the letters of ''B'' and conversely any map from ''B'' to ''M'' extends to a morphism. A morphism is '''non-erasing'''<ref name=Lot7>{{harvtxt|Lothaire|1997|p=7}}</ref> or '''continuous'''<ref name=Sak25>{{harvtxt|Sakarovitch|2009|p=25}}</ref> if no letter of ''B'' maps to ι and '''trivial''' if every letter of ''B'' maps to ι.<ref name=Lot164>{{harvtxt|Lothaire|1997|p=164}}</ref>
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| A morphism ''f'' from a free monoid ''B''<sup>∗</sup> to a free monoid ''A''<sup>∗</sup> is '''total''' if every letter of ''A'' occurs in some word in the image of ''f''; '''cyclic'''<ref name=Lot164/> or '''periodic'''<ref name=Sal77>Salomaa (1981) p.77</ref> if the image of ''f'' is contained in{{clarify|reason=The image of f is a set of strings, while w° is a string. None of the meanings of 'is contained in' fits, i.e. neither 'is an element of', nor 'is a subset of', nor 'is a substring of'. Probably, every value f(v) is required to be a substring of w°.|date=August 2013}} ''w''<sup>∗</sup> for some word ''w'' of ''A''<sup>∗</sup>. A morphism ''f'' is '''''k''-uniform''' if the length |''f''(''a'')| is constant and equal to ''k'' for all ''a'' in ''A''.<ref name=ApCoW522>{{harvtxt|Lothaire|2005|p=522}}</ref><ref name=BR103>{{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 | page=103 }}</ref> A 1-uniform morphism is '''strictly alphabetic'''<ref name=Sak25/> or a '''coding'''.<ref name=AS9>{{harvtxt|Allouche|Shallit|2003|p=9}}</ref>
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| A morphism ''f'' from a free monoid ''B''<sup>∗</sup> to a free monoid ''A''<sup>∗</sup> is '''simplifiable''' if there is an alphabet ''C'' of cardinality less than that of ''B'' such the morphism ''f'' factors through ''C''<sup>∗</sup>;{{clarify|reason=Please give a definition of 'factors through' or a link to it.|date=August 2013}} otherwise ''f'' is '''elementary'''. The morphism ''f'' is called a '''code''' if the image of the alphabet ''B'' under ''f'' is a code: every elementary morphism is a code.<ref name=Sal72>Salomaa (1981) p.72</ref>
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| ===Test sets===
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| For ''L'' a subset of ''B''<sup>∗</sup>, a finite subset ''T'' of ''L'' is a ''test set'' for ''L'' if morphisms ''f'' and ''g'' on ''B''<sup>∗</sup> agree on ''L'' if and only if they agree on ''T''. The '''Ehrenfeucht conjecture''' is that any subset ''L'' has a test set:<ref name=Lot1789>{{harvtxt|Lothaire|1997|pp=178–179}}</ref> it has been proved<ref name=LotII451>{{harvtxt|Lothaire|2011|p=451}}</ref> independently by Albert and Lawrence; McNaughton; and Guba. The proofs rely on [[Hilbert's basis theorem]].<ref>{{cite journal | first=A. | last=Salomaa | authorlink=Arto Salomaa | title=The Ehrenfeucht conjecture: A proof for language theorists | journal=Bulletin of the EATCS | year=1985 | number=27 | date=October 1985 | pages=71–82 }}</ref>
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| ==Endomorphisms==
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| An '''[[endomorphism]]''' of ''A''<sup>∗</sup> is a morphism from ''A''<sup>∗</sup> to itself.<ref name=LotII450>{{harvtxt|Lothaire|2011|p=450}}</ref> The [[identity map]] ''I'' is an endomorphism of ''A''<sup>∗</sup>, and the endomorphisms form a [[monoid]] under [[composition of functions]].
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| An endomorphism ''f'' is '''prolongable''' if there is a letter ''a'' such that ''f''(''a'') = ''as'' for a non-empty string ''s''.<ref name=AS10>Allouche & Shallit (2003) p.10</ref>
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| ===String projection===
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| The operation of [[String_operations#String_projection|string projection]] is an endomorphism. That is, given a letter ''a'' ∈ Σ and a string ''s'' ∈ Σ<sup>∗</sup>, the string projection ''p''<sub>a</sub>(''s'') removes every occurrence of ''a'' from ''s''; it is formally defined by
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| :<math>p_a(s) = \begin{cases}
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| \varepsilon & \text{if } s=\varepsilon, \text{ the empty string} \\
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| p_a(t) & \text{if } s=ta \\
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| p_a(t)b & \text{if } s=tb \text{ and } b\ne a.
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| \end{cases}</math>
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| Note that string projection is well-defined even if the rank of the monoid is infinite, as the above recursive definition works for all strings of finite length. String projection is a [[morphism]] in the category of free monoids, so that
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| :<math>p_a\left(\Sigma^*\right)= \left(\Sigma-a\right)^*</math> | |
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| where <math>p_a\left(\Sigma^*\right)</math> is understood to be the free monoid of all finite strings that don't contain the letter ''a''. The identity morphism is <math>p_\varepsilon</math>,{{clarify|reason=Up to here, string projection has been defined only for single letters, not for arbitrary strings. However, the empty string ε is not a letter.|date=August 2013}} as clearly <math>p_\varepsilon(s)=s</math> for all strings ''s''. Of course, it commutes with the operation of string concatenation, so that <math>p_a(st)=p_a(s)p_a(t)</math> for all strings ''s'' and ''t''. There are many right inverses to string projection, and thus it is a [[split epimorphism]].
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| String projection is commutative, as clearly
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| :<math>p_a(p_b(s))=p_b(p_a(s)).</math>
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| For free monoids of finite rank, this follows from the fact that free monoids of the same rank are isomorphic, as projection reduces the rank of the monoid by one.
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| String projection is [[idempotent]], as
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| :<math>p_a(p_a(s))=p_a(s)</math>
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| for all strings ''s''. Thus, projection is an idempotent, commutative operation, and so it forms a bounded [[semilattice]] or a commutative [[band (algebra)|band]].
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| ===Sturmian endomorphisms===
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| An endomorphism of the free monoid ''B''<sup>∗</sup> on a 2-letter alphabet ''B'' is '''Sturmian''' if it maps every [[Sturmian word]] to a Sturmian word<ref name=LotII83>{{harvtxt|Lothaire|2011|p=83}}</ref><ref name=PF197>{{harvtxt|Pytheas Fogg|2002|p=197}}</ref> and '''locally Sturmian''' if it maps some Sturmian word to a Sturmian word.<ref name=LotII85>{{harvtxt|Lothaire|2011|p=85}}</ref> The Sturmian endomorphisms form a submonoid of the monoid of endomorphisms of ''B''<sup>∗</sup>.<ref name=LotII83/>
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| Define endomorphisms φ and ψ of ''B''<sup>∗</sup>, where ''B'' = {0,1}, by φ(0) = 01, φ(1) = 0 and ψ(0) = 10, ψ(1) = 0. Then ''I'', φ and ψ are Sturmian,<ref name=LotII84>{{harvtxt|Lothaire|2011|p=84}}</ref> and the Sturmian endomorphisms of ''B''<sup>∗</sup> are precisely those endomorphisms in the submonoid of the endomorphism monoid generated by {''I'',φ,ψ}.<ref name=BS1994>{{cite journal | last1=Berstel | first1=J. | last2=Séébold | first2=P. | title=A remark on morphic Sturmian words | journal=RAIRO, Inform. Théor. Appl. 2| volume=8 | number=3-4 | pages=255–263 | year=1994 | issn=0988-3754 | zbl=0883.68104 }}</ref><ref name=LotII85/><ref name=PF197/>
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| A primitive substitution is Sturmian if the image of the word 10010010100101 is balanced.{{clarify|reason=Define 'primitive substitution' and 'balanced'; indicate whether the notion of a 'sturmian substitution' is subsumed by the definition of a 'sturmian endomorphism' or whether is something different.|date=August 2013}}<ref name=BS1993>{{citation | chapter=A characterization of Sturmian morphisms | first1=Jean | last1=Berstel | first2=Patrice | last2=Séébold | title=Mathematical Foundations of Computer Science 1993. 18th International Symposium, MFCS'93 Gdańsk, Poland, August 30–September 3, 1993 Proceedings | editor1-first=Andrzej M. | editor1-last=Borzyszkowski | editor2-first=Stefan | editor2-last=Sokołowski | pages=281–290 | year=1993 | doi=10.1007/3-540-57182-5_20 | isbn=978-3-540-57182-7 | series=Lecture Notes in Computer Science | volume=711 | zbl=0925.11026 }}</ref><ref name=PF197/>
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| ==The free commutative monoid==
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| Given a set ''A'', the '''free [[commutative monoid]]''' on ''A'' is the set of all finite [[multiset]]s with elements drawn from ''A'', with the monoid operation being multiset sum and the monoid unit being the empty multiset.
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| For example, if ''A'' = {''a'', ''b'', ''c''}, elements of the free commutative monoid on ''A'' are of the form
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| :{ε, ''a'', ''ab'', ''a''<sup>2</sup>''b'', ''ab''<sup>3</sup>''c''<sup>4</sup>, ...}.
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| The [[fundamental theorem of arithmetic]] states that the monoid of positive integers under multiplication is a free commutative monoid on an infinite set of generators, the [[prime number]]s.
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| The '''free commutative semigroup''' is the subset of the free commutative monoid which contains all multisets with elements drawn from ''A'' except the empty multiset.
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| ==Generalization==
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| The [[free partially commutative monoid]], or ''[[trace monoid]]'', is a generalization that encompasses both the free and free commutative monoids as instances. This generalization finds applications in [[combinatorics]] and in the study of [[Parallel computing|parallelism]] in [[computer science]].
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| ==Free monoids and computing==
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| The free monoid on a set ''A'' corresponds to [[List (computing)|lists]] of elements from ''A'' with concatenation as the binary operation. A [[monoid homomorphism]] from the free monoid to any other monoid (''M'',•) is a function ''f'' such that
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| * ''f''(''x''<sub>1</sub>…''x''<sub>''n''</sub>) = ''f''(''x''<sub>1</sub>) • … • ''f''(''x''<sub>''n''</sub>)
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| * ''f''() = ''e''
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| where ''e'' is the identity on ''M''. Computationally, every such homomorphism corresponds to a [[Map (higher-order function)|map]] operation applying ''f'' to all the elements of a list, followed by a [[Fold (higher-order function)|fold]] operation which combines the results using the binary operator •. This [[squiggol|computational paradigm]] (which can be generalised to non-associative binary operators) has inspired the [[MapReduce]] software framework.
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| ==See also==
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| * [[String operations]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{citation | last1 = Allouche | first1 = Jean-Paul | last2 = Shallit | first2 = Jeffrey | author2-link = Jeffrey Shallit | isbn = 978-0-521-82332-6 | publisher = [[Cambridge University Press]] | title = Automatic Sequences: Theory, Applications, Generalizations | year = 2003 | zbl=1086.11015 }}
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| * {{citation | last1=Berstel | first1=Jean | last2=Perrin | first2=Dominique | last3=Reutenauer | first3=Christophe | title=Codes and automata | series=Encyclopedia of Mathematics and its Applications | volume=129 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2010 | isbn=978-0-521-88831-8 | zbl=1187.94001 }}
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| *{{citation | last=Lothaire | first=M. | authorlink=M. Lothaire | others=Contributors: Perrin, D.; Reutenauer, C.; Berstel, J.; Pin, J. E.; Pirillo, G.; Foata, D.; Sakarovitch, J.; Simon, I.; Schützenberger, M. P.; Choffrut, C.; Cori, R. Series editors: Lyndon, Roger; Rota, Gian-Carlo. Foreword by Roger Lyndon | title=Combinatorics on words | edition=2nd | series=Cambridge Mathematical Library | volume=17 | publisher=[[Cambridge University Press]] | year=1997 | doi = 10.1017/CBO9780511566097 | isbn=0-521-59924-5 | mr = 1475463 | zbl=0874.20040 }}
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| * {{citation | last=Lothaire | first=M. | authorlink=M. Lothaire | title=Algebraic combinatorics on words | others=With preface by Jean Berstel and Dominique Perrin | edition=Reprint of the 2002 hardback | series=Encyclopedia of Mathematics and Its Applications | volume=90| publisher=[[Cambridge University Press]] | year=2011 | isbn=978-0-521-18071-9 | zbl=1221.68183 }}
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| *{{citation | 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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| * {{citation | last=Pytheas Fogg | first=N. | others=Editors Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. | title=Substitutions in dynamics, arithmetics and combinatorics | series=Lecture Notes in Mathematics | volume=1794 | location=Berlin | publisher=[[Springer-Verlag]] | year=2002 | isbn=3-540-44141-7 | zbl=1014.11015 }}
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| * {{citation | 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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| * {{citation | first=Arto | last=Salomaa | authorlink=Arto Salomaa | title=Jewels of Formal Language Theory | publisher=Pitman Publishing | isbn=0-273-08522-0 | year=1981 | zbl=0487.68064 }}
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| [[Category:Semigroup theory]]
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| [[Category:Formal languages]]
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| [[Category:Free algebraic structures]]
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| [[Category:Combinatorics on words]]
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