Can BLS aggregate signatures be merged?

Question

In some non-interactive, pairing-friendly signature scheme, such as BLS12-381, is it possible to merge partially-overlapping aggregate signatures?

For example, say you have two aggregate signatures, $X = a + b + c$ and $Y = b + c + d$, and the (unique) public keys and messages for individual signatures $a$ through $d$, but not the individual signatures anymore.

Can you make a "merged" aggregate signature $Z = a + b + c + d$?

Can you make a "combined" aggregate signature $W = a + 2b + 2c + d$ that can still be verified, perhaps by keeping track of the individual multipliers?

Answer 1

You cannot create a verifiable $Z$ but you can create a verifiable $W$ but you have to add public keys multiple times as well. So $W$ can be verified using a key $PK_a + 2*PK_b + 2*PK_c + PK_d$ if you had the same message or doing having multiple pairings for each repeated key in case of distinct messages