Epimorphisms in Cylindric Algebras and Definability in Finite Variable Logic

The main result gives a sufficient condition for a class K of finite dimensional cylindric algebras to have the property that not every epimorphism in K is surjective. In particular, not all epimorphisms are surjective in the classes CAn of n-dimensional cylindric algebras and the class of representable algebras in CAn for finite n > 1, solving Problem 10 of [28] for finite n. By a result of Németi, this shows that the Beth-definability property fails for the finite-variable fragments of first order logic as long as the number n of variables available is > 1 and we allow models of size ≥ n + 2, but holds if we allow only models of size ≤ n + 1. We also use our results in the present paper to prove that several results in the literature concerning injective algebras and definability of polyadic operations in CAn are best possible. We raise several open problems. §0. INTRODUCTION AND THE MAIN RESULTS In algebra, the properties of epimorphisms (in the categorial sense) being surjective and the amalgamation property in a class of algebras are well investigated, see e.g. [1] and [37]. In algebraic logic these properties turn out to be the algebraic equivalents of Beth’s definability property and Craig’s interpolation property, respectively (of the logic under algebraization), see [27, sec 5.6 Thm 5.6.10] for the first equivalence and [57, Thm 1.2.8] for the second equivalence. We understand Beth’s and Craig’s properties of abstract (or general) logics in the abstract model theoretic sense, cf. Barwise-Feferman [13, p.32, Def 1.2.4], or [9, sec 6]. The equivalence result concerning Craig’s property can be traced back to Daigneault [19] in the context of polyadic algebras. Pigozzi [57] is a milestone for working out such equivalences for cylindric algebras, an alternative equational formalism of first order logic. The corresponding algebraic question (amalgamation property of cylindric algebras) was largely settled by Comer [16] (the finite dimensional case) and Pigozzi [57] (the infinite dimensional case). The equivalence between amalgamation and interpolation is studied in more general contexts in [40],[41] and [42]. The