prefix set

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Let X be a set, and w∈X∗ be a word, i.e. an element of the free monoid on X. A word v∈X∗ is called prefix of w when a second word z∈X∗ exists such that x=vz. A proper prefix of a word u is a prefix v of u not equal to u (sometimes v is required to be non-empty).
Note that the empty word ε and w are prefix of w, and a proper prefix of w if w is non-empty.
The prefix set of w is the set pref(w) of prefixes of w, i.e. if w=w1w2...wn with wj∈X for each j∈1,...,n we have

pref(w)=ε,w1,w1w2,...,w1w2...wn−1,w.

Some closely related concepts are:
1. A set of words is prefix closed if for every word in the set, any of its prefix is also in the set.
2. The prefix closure of a set S is the smallest prefix closed set containing S, or, equivalently, the union of the prefix sets of words in S.
3. A set S is prefix free if for any word in S, no proper prefixes of the word are in S.

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