Amalgamation in finite dimensional cylindric algebras

Abstract

For every finite n > 1, the embedding property fails in the class of all n-dimensional cylindric type algebras which satisfy the following. Their boolean reducts are boolean algebras and two of the cylindrifications are normal, additive and commute. This result also holds for all subclasses containing the representable n-dimensional cylindric algebras. This considerably strengthens a result of S. Comer on CA n and provides a strong counterexample for interpolation in finite variable fragments of first order logic. We provide a new modern proof, using an argument inspired by modal logic.S. Comer showed that for finite n larger than 1, the embedding property fails in any class of cylindric type algebras of dimension n lying between RCA n and CA n [3]. On the other hand, the proof method employed by I. Németi in [9] can be used to show that for any α, the class NCA α (defined by all CA α -axioms except commutativity of the cylindrifications) and its subclass D α of cylindric-relativised set algebras have the strong amalgamation property. In this last class, commutativity only holds in the restricted sense of c i (c j . So some combination of the NCA axioms together with some commutativity of the cylindrifications can be seen as the cause of the failure of the embedding property. It is a natural question then to ask how much commutativity and how much of the other axioms is really needed to obtain Comer's result. It turns out that for every finite n, we only need to ask that the first two cylindrifications are normal, additive and commute in only one direction. So we find a unique reason for the failure of the embedding property for every n. This considerably strengthens Comer's result. The difference lies not only in the required properties of the cylindrifications (e.g., in CA n , they are complemented closure operators, in our case they are just operations), but in particular that for any finite n, the failure of the embedding property is caused by the behaviour of the same two cylindrifications. We provide a new modern proof, using an argument inspired by modal logic. These algebraic results have -by the correspondence theorems developed in [10]-a direct logical meaning: they provide a strong counterexample for interpolation in first order logic with only finitely many variables.We recall the definitions of the above mentioned classes of cylindric type algebras and their operations from [6]. An algebra A= (A, ∧, ∨, −, 0, 1, c κ , d κλ ) κ,λ<α is a cylindric

On amalgamation of reducts of polyadic algebras

“…Fragments of arrow logic have been studied quite extensively lately; their algebraic counterpart turns out to be reducts and variants of Tarski's relation algebras. Marx's main results in [46], and other results concerning amalgamation in cylindric-like algebras of relations are streamlined in [47] and [48]. 27 As previously noted, it is not known, at least to the author, whether there exists a variety V of CA-like algebras of relations that separates the notions of strong and super amalgamation.…”

“…Since the story was often told elsewhere (see e.g. [56], [57], [13], [46], [48], [45], [41] and [42]), in Section 3 we only briefly discuss the logical implications mentioned in the abstract (cf. item (1)).…”

“…The proofs of the two above mentioned Lemmas are generalizations of the techniques of Németi, developed in [52] for proving the strong amalgamation property for various subclasses of boolean algebras with operators and relativized versions of set algebras such as for example D α and NCA α . The reader is referred to [48] for the definition of such subclasses. The latter is the class of non-commutative cylindric algebras.…”

Amalgamation, interpolation and epimorphisms in algebraic logic

“…We will give a detailed account of this at the end of Section 3. In Section 4 we survey results concerning amalgamation for other cylindric-like algebras of relations, such as diagonal-free reducts of CA's, [33], [24], Pinter's substitution algebras [32], Halmos' polyadic algebras [15], and various reducts thereof, such as Halmos' quasipolyadic (equality) algebras and those reducts investigated in [34] and [39]. At the end of Section 4 we also summarize (both positive and negative) results on the amalgamation for the above mentioned relatives of CA's in tabular form analogous to Table 1.…”

Varieties of two-dimensional cylindric algebras.¶Part I: Diagonal-free case

“…It was already noticed by Tarski that Df 2 is not locally finite (see, e.g., Henkin, Monk and Tarski [9], Halmos [8], and below). That Df 2 does not have the amalgamation property was first noticed by Comer [4] (see also Sain [18] and Marx [16]). On the other hand, it is known that Df 2 is finitely approximable.…”