Private set intersection using homomorphic encryption

I have a question regarding the scheme below. PSI using homomorphic encryption appears to subtract the plaintext into the ciphertext. I understand that homomorphism is maintained in operations between ciphertexts. Is homomorphism maintained even if calculated as below?

If 3. (b) operation is correct, c=[c0,c1] and x is a single element. How 3.(b) operation can be possible?

• It would help to know what paper this is from. However, my guess is that it's just sloppy notation to refer to "homomorphically compute the polynomial $r_i\prod_{x \in X} (m_i - x)$", where $m_i$ is the plaintext message encoded in $c_i$. Specifically, note that the sender and receiver "agree on an FHE scheme" so there is no guarantee that the ciphertext is even of the form $c=[c0,c1]$ or that the arithmetic operations on the ciphertexts translate to the plaintext space. Jan 29 at 9:17
• Alternatively, assuming a specific scheme (e.g., BGV), $x$ can first be encrypted by the sender using the public key (to have the form $[c_0, c_1]$) and then used to compute the operations as described. However, this would assume the arithmetic operations translate to plaintext space. Jan 29 at 9:20