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prefix set (Definition)

Let $ X$ be a set, and $ w\in X^*$ be a word, i.e. an element of the free monoid on $ X$. A word $ v\in X^*$ is called prefix of $ w$ when a second word $ z\in X^*$ exists such that $ x=vz$.

Note that the empty word $ \varepsilon$ and $ w$ are prefix of $ w$. The prefix set of $ w$ is the set $ \mathrm{pref}(w)$ of prefixes of $ w$, i.e. if $ w=w_1w_2...w_n$ with $ w_j\in X$ for each $ j\in\{1,...,n\}$ we have

$\displaystyle \mathrm{pref}(w)=\{\varepsilon,\ w_1,\ w_1w_2,\ ... ,\ w_1w_2...w_{n-1},\ w\}.$



"prefix set" is owned by Mazzu.
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Also defines:  prefix, prefix set
Keywords:  free semigroup, word
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Cross-references: empty word, free monoid, word
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This is version 2 of prefix set, born on 2006-08-25, modified 2006-08-25.
Object id is 8292, canonical name is PrefixSet.
Accessed 2519 times total.

Classification:
AMS MSC: 20M05 (Group theory and generalizations :: Semigroups :: Free semigroups, generators and relations, word problems)
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