Comparable or partially homomorphic public key derivation for signatures?
If I understand you correctly, you want that given $\mathit{pk}, \mathit{pk}'$, one can publicly decide whether or not $\mathit{sk} &lt; \mathit{sk}'$ (lexicographically) for the corresponding secret keys. 

Such a signature scheme would inherently be insecure. 
This is because given your public key $\mathit{pk}^*$, anyone would be able to run a binary search to find the underlying secret key $\mathit{sk}^*$. 

For example, the attacker could choose $\mathit{sk} = 10000000$, compute the corresponding $\mathit{pk}$ and then run the lexicographic comparison on $\mathit{pk}$ and $\mathit{pk}^*$ to find out whether or not your key $\mathit{sk}^*$ starts with a 1. If so, checking whether $\mathit{sk}^*$ is still lexicographically larger than $11000000$ can be done similarly and reveals the next bit, etc.

After $n$ comparisons (assuming n-bit secrets), the secret key is revealed. Of course, this also generalizes to larger alphabets.