prefix set
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 $x=vz$. A proper prefix of a word u is a prefix v of u not equal to u (sometimes v is required to be nonempty).
Note that the empty word^{} $\epsilon $ and $w$ are prefix of $w$, and a proper prefix of $w$ if $w$ is nonempty.
The prefix set of $w$ is the set $\mathrm{pref}(w)$ of prefixes of $w$, i.e. if $w={w}_{1}{w}_{2}\mathrm{\dots}{w}_{n}$ with ${w}_{j}\in X$ for each $j\in \{1,\mathrm{\dots},n\}$ we have
$$\mathrm{pref}(w)=\{\epsilon ,{w}_{1},{w}_{1}{w}_{2},\mathrm{\dots},{w}_{1}{w}_{2}\mathrm{\dots}{w}_{n1},w\}.$$ 
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.
Title  prefix set 
Canonical name  PrefixSet 
Date of creation  20130322 16:11:56 
Last modified on  20130322 16:11:56 
Owner  Mazzu (14365) 
Last modified by  Mazzu (14365) 
Numerical id  6 
Author  Mazzu (14365) 
Entry type  Definition 
Classification  msc 20M05 
Defines  prefix 
Defines  prefix set 
Defines  proper prefix 
Defines  prefix closed 
Defines  prefix closure 
Defines  prefix free 