On the algebraization of Henkin-type second-order logic

There is an extensive literature related to the algebraization of first-order logic. But the algebraization of full second-order logic, or Henkin-type second-order logic, has hardly been researched. The question arises: what kind of set algebra is the algebraic version of a Henkin-type model of second-order logic? The question is investigated within the framework of the theory of cylindric algebras. The answer is: a kind of cylindric-relativized diagonal restricted set algebra. And the class of the subdirect products of these set algebras is the algebraization of Henkin-type second-order logic. It is proved that the algebraization of a complete calculus of the Henkin-type second-order logic is a class of a kind of diagonal restricted cylindric algebras. Furthermore, the connection with the non-standard enlargements of standard complete second-order structures is investigated.