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

Vaught’s Conjecture states that if Σ is a complete first order theory in a countable language such that Σ has uncountably many pairwise non-isomorphic countably infinite models, then Σ has 2א0 many pairwise non-isomorphic countably infinite models. Continuing investigations initiated in [17], we apply methods of algebraic logic to study some variants of Vaught’s conjecture. More concretely, let S ⊆ ωω be a σ-compact monoid. We prove, among other things, that if a complete first order theory Σ has at least א1 many countable models which cannot be elementarily embedded into each other by elements of S, then, in fact, Σ 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 quasipolyadic algebras (for details see [9] and [10]) and topological properties of the Stone spaces of these algebras. AMS Subject Classification: Primary 03C45, 03G15; Secondary 03C30.