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A (linear) sequence of letters (cf. [[Letter|Letter]]) from some [[Alphabet|alphabet]]. For example, the series of symbols  "wordinanalphabet"  is a word in any alphabet containing the letters i, w, o, r, d, n, a, l, p, h, b, e, t. For convenience, one also allows the empty word, that is, the word containing no letters. It is a word in any alphabet.
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A (linear) sequence of letters (cf. [[Letter]]) from some [[alphabet]]. For example, the series of symbols  "wordinanalphabet"  is a word in any alphabet containing the letters i, w, o, r, d, n, a, l, p, h, b, e, t. For convenience, one also allows the empty word, that is, the word containing no letters. It is a word in any alphabet.
  
More precisely, one can use an inductive characterization of a word, whereby the words in an alphabet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098130/w0981301.png" /> are defined as the objects obtained by the following generating process: a) the empty word is a word in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098130/w0981302.png" />; b) if an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098130/w0981303.png" /> is a word in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098130/w0981304.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098130/w0981305.png" /> is a letter of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098130/w0981306.png" />, then the object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098130/w0981307.png" /> is also a word in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098130/w0981308.png" />. This characterization of words makes it possible to apply inductive arguments in proving universally true statements about the words in a given alphabet.
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More precisely, one can use an inductive characterization of a word, whereby the words in an alphabet $A$ are defined as the objects obtained by the following generating process: a) the empty word $\epsilon$ is a word in $A$; b) if an object $P$ is a word in $A$ and $\xi$ is a letter of $A$, then the object $P \xi$ is also a word in $A$. This characterization of words makes it possible to apply inductive arguments in proving universally true statements about the words in a given alphabet.
  
A word is a fairly general type of [[Constructive object|constructive object]], and because of this, the notion of a word plays an important role in constructive mathematics. The concept of a word is also widely used in algebra, mathematical linguistics and elsewhere.
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A word is a fairly general type of [[constructive object]], and because of this, the notion of a word plays an important role in constructive mathematics. The concept of a word is also widely used in algebra, mathematical linguistics and elsewhere.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Markov,  "Theory of algorithms" , Israel Program Sci. Transl.  (1961)  (Translated from Russian)  (Also: Trudy Mat. Inst. Steklov. 42 (1954))</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Markov,  N.M. [N.M. Nagornyi] Nagorny,  "The theory of algorithms" , Kluwer  (1988)  (Translated from Russian)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Markov,  "Theory of algorithms" , Israel Program Sci. Transl.  (1961)  (Translated from Russian)  (Also: Trudy Mat. Inst. Steklov. 42 (1954))</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Markov,  N.M. [N.M. Nagornyi] Nagorny,  "The theory of algorithms" , Kluwer  (1988)  (Translated from Russian)</TD></TR>
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</table>
  
  
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In algebra, words normally consist of letters and operation symbols, as  "x+y-z" .
 
In algebra, words normally consist of letters and operation symbols, as  "x+y-z" .
  
The length of a word is defined inductively: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098130/w0981309.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098130/w09813010.png" />.
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The length of a word is defined inductively: $\ell(\epsilon)=0$, $\ell(P\xi) = \ell(P) + 1$.
  
 
Under concatenation
 
Under concatenation
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$$
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(a_1 \cdots a_m,b_1 \cdots b_n) \mapsto a_1\cdots a_m b_1 \cdots b_n
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$$
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the set $\Omega(A)$ of all words in an alphabet $A$ becomes an associative [[monoid]]. The empty word is the unit element. This is the [[free monoid]] over $A$. It satisfies the freeness property: For every monoid $M$ and mapping of sets $\phi : A \rightarrow M$ there is a unique morphism of monoids $\tilde\phi : \Omega(A) \rightarrow M$ extending $\phi$. Here, $A$ is identified with the set of words of length $1$ in $\Omega(A)$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098130/w09813011.png" /></td> </tr></table>
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====References====
 +
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  R.C. Lyndon,  P.E. Schupp,  "Combinatorial group theory" , Springer  (1977)</TD></TR>
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</table>
  
the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098130/w09813012.png" /> of all words in an alphabet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098130/w09813013.png" /> becomes an associative [[Monoid|monoid]]. The empty word is the unit element. This is the free monoid over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098130/w09813014.png" />. It satisfies the freeness property: For every monoid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098130/w09813015.png" /> and mapping of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098130/w09813016.png" /> there is a unique morphism of monoids <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098130/w09813017.png" /> extending <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098130/w09813018.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098130/w09813019.png" /> is identified with the set of words of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098130/w09813020.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098130/w09813021.png" />.
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{{TEX|done}}
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.C. Lyndon,  P.E. Schupp,  "Combinatorial group theory" , Springer  (1977)</TD></TR></table>
 

Revision as of 20:26, 8 December 2015

A (linear) sequence of letters (cf. Letter) from some alphabet. For example, the series of symbols "wordinanalphabet" is a word in any alphabet containing the letters i, w, o, r, d, n, a, l, p, h, b, e, t. For convenience, one also allows the empty word, that is, the word containing no letters. It is a word in any alphabet.

More precisely, one can use an inductive characterization of a word, whereby the words in an alphabet $A$ are defined as the objects obtained by the following generating process: a) the empty word $\epsilon$ is a word in $A$; b) if an object $P$ is a word in $A$ and $\xi$ is a letter of $A$, then the object $P \xi$ is also a word in $A$. This characterization of words makes it possible to apply inductive arguments in proving universally true statements about the words in a given alphabet.

A word is a fairly general type of constructive object, and because of this, the notion of a word plays an important role in constructive mathematics. The concept of a word is also widely used in algebra, mathematical linguistics and elsewhere.

References

[1] A.A. Markov, "Theory of algorithms" , Israel Program Sci. Transl. (1961) (Translated from Russian) (Also: Trudy Mat. Inst. Steklov. 42 (1954))
[2] A.A. Markov, N.M. [N.M. Nagornyi] Nagorny, "The theory of algorithms" , Kluwer (1988) (Translated from Russian)


Comments

In algebra, words normally consist of letters and operation symbols, as "x+y-z" .

The length of a word is defined inductively: $\ell(\epsilon)=0$, $\ell(P\xi) = \ell(P) + 1$.

Under concatenation $$ (a_1 \cdots a_m,b_1 \cdots b_n) \mapsto a_1\cdots a_m b_1 \cdots b_n $$ the set $\Omega(A)$ of all words in an alphabet $A$ becomes an associative monoid. The empty word is the unit element. This is the free monoid over $A$. It satisfies the freeness property: For every monoid $M$ and mapping of sets $\phi : A \rightarrow M$ there is a unique morphism of monoids $\tilde\phi : \Omega(A) \rightarrow M$ extending $\phi$. Here, $A$ is identified with the set of words of length $1$ in $\Omega(A)$.

References

[a1] R.C. Lyndon, P.E. Schupp, "Combinatorial group theory" , Springer (1977)
How to Cite This Entry:
Word. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Word&oldid=36870
This article was adapted from an original article by N.M. Nagornyi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article