The class $\textsf{Cs}_{\omega}^{reg}\cap \textsf{Lf}_{\omega}$ of locally finite and regular cylindric set algebras (of dimension $\omega$) can be seen as the algebraic counterpart of first-order models (details below).
I also know that we can associate, to each model, a representable cylindric algebra in $RCA_{\alpha}$, where $\alpha$ denotes the number of variables in our signature.
- What is the relation between $\textsf{RCA}_{\alpha}$ and $\textsf{Cs}_{\alpha}^{reg}\cap \textsf{Lf}_{\alpha}$, where $\alpha$ is some at most countable ordinal?
In particular, I am interested in the equational definability of the regularity property. The definition of regularity for cylindric set algebras is always given by explicit reference to tuples from the underlying `domain set' -- the set of tuples $U^{\alpha}$ which serves as basis of the given cylindric algebra (i.e. the cylindric algebra is an algebra over $\mathcal{P}(U^{\alpha})$).
Is it possible to define regularity using only the language of cylindric algebras?
Is the class of regular, locally finite cylindric set algebras equationally definable ?
A more general question: suppose we want to give a complete semantics for first-order logic in terms of cylindric algebras. That is, we want each model to be given by a pair $(\mathfrak{A}, [\cdot])$ where $\mathfrak{A}$ is a cylindric algebra and $[\cdot]$ a valuation mapping each formula to an element $a\in\mathfrak{A}$ in the usual way. Can we do so in such a way that the required class of cylindric algebras is a variety? Can we characterise the class of all such pairs $(\mathfrak{A}, [\cdot])$ in an abstract way, without mention of the underlying set of tuples $U^{\alpha}$ representing the domain of some 'corresponing' first-order model?
What are the natural classes of algebras that could serve as candidates for this?
(if not, can we do so for a first-order language without equality? or with finitely many variables?)
Some more details:
Fix a signature (first-order language).
To each first-order model $\mathfrak{M}$ there corresponds an $\omega$-dimensional cylindric set algebra $\textsf{Cs}^{\mathfrak{M}}$, obtained by mapping each formula to the set of tuples in $\textsf{dom}(\mathfrak{M})$ that satisfy it. Such an algebra $\textsf{Cs}^{\mathfrak{M}}$ is always locally finite and regular. Conversely, each locally finite and regular cylindric set algebra $\textsf{A}$ has a corresponding model $\mathfrak{M}$ such that $\textsf{A}=\textsf{Cs}^{\mathfrak{M}}$.