# alphabet

An alphabet $\Sigma$ is a nonempty finite set  such that every string formed by elements of $\Sigma$ can be decomposed uniquely into elements of $\Sigma$.

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. $\{\mathbb{C}a,\ddot{n}a,{\rm d},a\}$ is a valid alphabet, because there is only one way to fully break up any given string formed from it.

If $\Sigma$ is our alphabet and $n\in\mathbb{Z}^{+}$, we define the following as the powers of $\Sigma$:

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

• $\Sigma^{n}=\{xy|x\in\Sigma,y\in\Sigma^{n-1}\}$ ($xy$ is the juxtaposition of $x$ and $y$)

So, $\Sigma^{n}$ is the set of all strings formed from $\Sigma$ of length $n$.

 Title alphabet Canonical name Alphabet Date of creation 2013-03-22 12:15:58 Last modified on 2013-03-22 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