On varieties of cylindric algebras with applications to logic

AbstractMnα, Mgα, and Bgα denote the classes of minimal, monadic-generated, and binary-generated cylindric algebras of dimension α respectively, and EqK denotes the equational theory of the class K of algebras. In Theorem 2, we characterize those classes K ⊆ Mgα, α > 2, for which EqK is recursively enumerable (r.e.). As a corollary we obtain that EqMnα is not r.e. iff α ⩾ ω, EqMgα is not r.e. iff α > 2, EqBgα is r.e. for α ⩾ ω and EqMnα = EqMgα iff (α = 0 or α ⩾ ω). These results solve Problems 4.11, 4.12 and the problem in item (1) on p. (ii) of the introduction of Part II of Henkin-Monk-Tarski [11] and continue the investigations started in Monk [22]. We discuss at length the logical meaning and consequences in the introduction and in Section 2. The proofs of the above results rely on the fact that the set of satisfiable Diophantine equations is not decidable. We also show that the equational theory of monadic-generated relation algebras is not r.e. Some further results can be found in Theorems 5 and 6: in Theorem 5 we give a single equation that characterizes being of characteristic 0 in Mgω, in Theorem 6 we investigate how big Mgα is. We do some investigations concerning the lattice of varieties of CAα's, α ⩾ ω