Free Boolean algebras with closure operators and a conjecture of Henkin , Monk , and Tarski ∗

We characterize the finite-dimensional elements of a free cylindric algebra. This solves Problem 2.10 in [Henkin, Monk, Tarski: Cylindric Algebras, North-Holland, 1971 and 1985]. We generalize the characterization to quasivarieties of Boolean algebras with operators in place of cylindric algebras. Free algebras play an important role in universal algebra, see e.g. AndrékaJónsson-Németi [2]. Free algebras are even more important in algebraic logic, because they give information on proof-theoretic properties of a logic. Cf. e.g. [7, §5.6], [4], [12], [8]. Cylindric algebras are special Boolean algebras with closure operators. Tarski proved that any element of a free cylindric algebra behaves the same way for all but finitely many of these operators: x is either closed for all but finitely many or only finitely many operators. See Theorem 2.6.23 in [6]. It remained open which elements of a free algebra are closed to all but finitely many, and which are closed only to finitely many. [6, 2.6.24] contains a conjecture for a characterization, but Henkin, Monk, and Tarski write there that they were unable to verify this conjecture. They also formulate the conjecture as Problem 2.10 in [6]. In this paper we solve Problem 2.10 of [6] by showing that their conjecture was right. We prove more: we give information about which element is closed under which operators, and also we generalize the result to quasi-varieties of Boolean algebras with operators. ∗Research supported by the Hungarian National Foundation for scientific research grants No T30314 and T23234.