1. It is important to remark that proper fork algebras are quasi-concrete structures, since, as was pointed out by Andréka and Németi in a private communication, concrete structures must be fully characterized by their underlying domain, which does not happen with proper fork algebras because of the (hidden) operation ⋆.
2. Along the next theorems, by S we denote the operation of taking subalgebras of a given class of algebras. P takes direct products of algebras in a given class, and I takes isomorphic copies.
3. Notice that this formula has four variables ranging over individuals.