User:Jshflynn/Definition:Language power

Definition
Let $\Sigma$ be an alphabet.

Let $V$ be a language over $\Sigma$.

Then the $n$th power of $V$, where $n \in \mathbb{N}_0$, is denoted $V^{n}$ and defined inductively:



V^{n} = \begin{cases} \{\lambda\} & \text{if }n=0 \\ V^{n-1} \circ_L V & \text{if }n > 0 \end{cases} $

Note
The notation is the same as that of the subalphabet power as the language power is an extension of the definition of subalphabet power and could only be defined after formal language was defined.