Hodge doesn’t state this explicitly as far as I can tell, but it’s implicit in Lemma 1.2.2 (p.7). If Y is empty and the language has no constants, then ∅ satisfies the conditions for 〈∅〉B. It contains cB for every constant c (since there are no constants), and is closed under the functions, since there are no n-tuples to apply the functions to.
Diagonal Argument 
Dear Mr Weiss.
I have seen an answer from you at https://math.stackexchange.com/questions/4683244/what-would-happen-in-predicate-logic-if-the-domain-of-discourse-was-allowed-to-b
Would you be so kind as to tell me where in Hodges’ book A Shorter Model Theory I can find the example you mention about the substructure ⟨Y⟩=∅?
Best whises
Hodge doesn’t state this explicitly as far as I can tell, but it’s implicit in Lemma 1.2.2 (p.7). If Y is empty and the language has no constants, then ∅ satisfies the conditions for 〈∅〉B. It contains cB for every constant c (since there are no constants), and is closed under the functions, since there are no n-tuples to apply the functions to.