Gábor Sági

On the Equational Theory of Representable Polyadic Equality Algebras

Among others we will prove that the equational theory of ω dimensional representable polyadic equality algebras (RPEA ω s) is not schema axiomatizable. This result is in interesting contrast with the Daigneault-Monk representation theorem, which states that the class of representable polyadic algebras is finite schema-axiomatizable (and hence the equational theory of this class is finite schema-axiomatizable. as well). We will also show that the complexity of the equational theory of RPEA ω is also extremely high in the recursion theoretic sense. Finally, comparing the present negative results with the positive results of Ildiko Sain and Viktor Gyuris [12], the following methodological conclusions will be drawn: The negative properties of polyadic (equality) algebras can be removed by switching from what we call the “polyadic algebraic paradigm” to the “cylindric algebraic paradigm”.

A Note on Algebras of Substitutions

We will study the class RSAα of α-dimensional representable substitution algebras. RSAα is a sub-reduct of the class of representable cylindric: algebras, and it was an open problem in Andréka [1] that whether RSAα can be finitely axiomatized. We will show, that the answer is positive. More concretely, we will prove, that RSAα is a finitely axiomatizable quasi-variety. The generated variety is also described. We note that RSAα is the algebraic counterpart of a certain proportional multimodal logic and it is related to a natural fragment of first order logic, as well.

Some variants of Vaught's conjecture from the perspective of algebraic logic

Vaught’s Conjecture states that if T is a complete first order theory in a countable language such that T has uncountably many pairwise non-isomorphic countably infinite models, then T has 2^ℵ_0 many pairwise non-isomorphic countably infinite models. Continuing investigations initiated in S´agi, we apply methods of algebraic logic to study some variants of Vaught’s conjecture. More concretely, let S be a σ-compact monoid of selfmaps of the the natural numbers. We prove, among other things, that if a complete first order theory T has at least ℵ1 many countable models that cannot be elementarily embedded into each other by elements of S, then, in fact, T has continuum many such models. We also study-related questions in the context of equality free logics and obtain similar results. Our proofs are based on the representation theory of cylindric and quasi-polyadic algebras (for details see Henkin, Monk and Tarski (cylindric Algebras Part 1 and Part 2)) and topological properties of the Stone spaces of these algebras.

On nonrepresentable G-polyadic algebras with representable cylindric reducts

In [10], Sayed Ahmed recently has shown that there exists an infinite dimensional non-representable quasi-polyadic equality algebra (QPEAω, for short) with a representable cylindric reduct. In this paper we continue related investigations and show that if G⊆ωω is a semigroup containing at least one constant function, then a wide class of representable cylindric algebras occur as the cylindric reduct of some non-representable G-PEAω. More concretely, we prove that if A is an ω-dimensional cylindric set algebra with an infinite base set, then there exists a non-representable G-PEAω whose cylindric reduct is representable and contains an isomorphic copy of A.

Ultraproducts and Higher Order Formulas

Which ultraproducts preserve the validity of formulas of higher order logics? To answer this question, we will introduce natural topologies (which we call ultratopologies) on ultraproducts. We will show, that ultraproducts preserving certain higher order formulas can be characterized in terms of these topologies. As an application of the above results, we provide a constructive, purely model theoretic characterization for classes definable by second order existential formulas.