Given an alphabet, say $ \Sigma = \{0,1\}$ , I can make a one-to-one mapping from all possible strings $ x \in \Sigma^*$ to $ \mathbb{N}$ . This could be done by ordering $ \Sigma^*$ lexicographically and assigning the $ i$ th string $ x_i$ to number $ i \in \mathbb{N}$ .

But given strings $ x_i,x_j \in \Sigma^*$ , is there any special mapping such that the concatenation operation $ f:\Sigma^* \rightarrow \Sigma^* | (x_i,x_j) \rightarrow x_ix_j$ is also related to the usual addition performed over the corresponding indices $ i,j \in \mathbb{N}$ to which $ x_i$ and $ x_j$ are mapped ?

For instance, if I assign the character $ \{1\}$ to the number $ 1$ , and string $ x$ is assigned the number $ 10$ , is there a mapping such that the string $ x1$ is assigned the number $ 11$ ? (i.e. $ 10 + 1$ )