Cylindric set algebras and related structures

Abstract

The abstract theory of cylindric algebras is extensively developed in the book [HMT] by the authors.Several kinds of special set algebras were mentioned, primarily for motivational purposes, in that book.

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

“…However, familiarity with cylindric and quasi-polyadic algebras may be an advantage of the reader in later sections. The definitions and propositions of this section, along with proofs, can be found in either [9] or [10].…”

“…Definition 1.1.1 of [9]). We do not recall these postulates, because we do not use their concrete forms below.…”

“…Let τ ∈ ω ω and for every i ∈ ω let's substitute v τ (i) for v i simultaneously. As explained in Definitions 1.11.9 and 1.11.13 of [9], these kind of generalized substitutions can be introduced in every Lf , the derived function is denoted by s + τ . Let U be a set.…”

“…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.…”

“…We prove, among other things, that if a complete first order theory Σ has at least ℵ 1 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. …”