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

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

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 $\Omega(A)$. The notations $A^*$ for the set $\Omega(A)$ of all words over the alphabet $A$ (free monoid) and $A^+$ for the set of all non-empty words over $A$ (free semi-group) are common.

A language over $A$ is a set of words over $A$. See Formal language and Formalized language.

An infinite word over $A$ is a sequence $x_i$ with each $x_i \in A$ indexed by $i$ running over the natural numbers (one-sided, superword) or integers (two-sided, bi-infinite word).

A factor of a word $w = (w_1,w_2,\ldots)$ is a consecutive subsequence of letters $(w_i,w_{i+1},\ldots,w_j)$.

How to Cite This Entry:
Word. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Word&oldid=39767
This article was adapted from an original article by N.M. Nagornyi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article