Is there a way to map the concatenation operation over strings to the addition operation over $mathbb{N}$

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. $10 + 1$)