Languages and Grammar

If $L$ is a language, then the product $L \cdot L$ is denoted by $L^2$. The language product $L^n$ for every $n \in \{0,1,2,…\} is defined as follows:

$$ \begin{aligned} L^0 &= \{\Lambda\} \\ L^n &= L \cdot L^{n-1}, \text{if} \ n > 0 \end{aligned} $$

Example. If $L = \{a,bb\}$ then the first few powers of $L$ are… $$ \begin{aligned} L^0 &= \{\Lambda\} \\ L^1 &= L = \{a,bb\} \\ L^2 &= L \cdot L = \{aa,abb,bba,bbbb\} \\ L^3 &= L \cdot L^2 = \{aaa,aabb,abba,abbbb,bbaa,bbabb,bbbba,bbbbbb\} \end{aligned} $$