# Word

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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 are defined as the objects obtained by the following generating process: a) the empty word is a word in ; b) if an object is a word in and is a letter of , then the object is also a word in . 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)

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

The length of a word is defined inductively: , .

Under concatenation

the set of all words in an alphabet becomes an associative monoid. The empty word is the unit element. This is the free monoid over . It satisfies the freeness property: For every monoid and mapping of sets there is a unique morphism of monoids extending . Here, is identified with the set of words of length in .

#### 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=16569
This article was adapted from an original article by N.M. Nagornyi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article