alphabet
An alphabet $\mathrm{\Sigma}$ is a nonempty finite set^{} such that every string formed by elements of $\mathrm{\Sigma}$ can be decomposed uniquely into elements of $\mathrm{\Sigma}$.
For example, $\{b,lo,g,bl,og\}$ is not a valid alphabet because the string $blog$ can be broken up in two ways: b lo g and bl og. $\{\u2102a,\ddot{n}a,\mathrm{d},a\}$ is a valid alphabet, because there is only one way to fully break up any given string formed from it.
If $\mathrm{\Sigma}$ is our alphabet and $n\in {\mathbb{Z}}^{+}$, we define the following as the powers of $\mathrm{\Sigma}$:

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${\mathrm{\Sigma}}^{0}=\lambda $, where $\lambda $ stands for the empty string.

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${\mathrm{\Sigma}}^{n}=\{xyx\in \mathrm{\Sigma},y\in {\mathrm{\Sigma}}^{n1}\}$ ($xy$ is the juxtaposition of $x$ and $y$)
So, ${\mathrm{\Sigma}}^{n}$ is the set of all strings formed from $\mathrm{\Sigma}$ of length $n$.
Title  alphabet 
Canonical name  Alphabet 
Date of creation  20130322 12:15:58 
Last modified on  20130322 12:15:58 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  7 
Author  mathcam (2727) 
Entry type  Definition 
Classification  msc 03C07 
Synonym  powers of an alphabet 
Related topic  KleeneStar 
Related topic  Substring 
Related topic  Language 
Related topic  HuffmanCoding 
Related topic  Word 