In their aim to algebraize the predicate calculus Tarski, Henkin and Thompson [7], [8] constructed the theory of cylindrical algebras, and Halmos [1] created the theory of polyadic algebras. Both these theories are based on the Boolean algebras, i.e. the sets of axioms of these theories enclose the axioms of the Boolean algebras. Our theory is a generalization of the theory of cylindrical algebras. Its axiom system consists of two groups of axioms. The first group comprises the axioms of the Boolean rings, and the second one describes the unary operations Dt and Q, i = 1, 2,... (b -> a), P2. (a-+b)-+((b->c)->(a->c)\ P3. ((a ~>b) -> a) -> a, P4. a a b -> a, P5. a a b -> b, P6. a -> (b -> a a b).