Space and time via Topological and Tense cylindric algebras

Abstract

Let $\alpha$ be an arbritary ordinal, and $2<n<\omega$. In \cite{3} accepted for publication in Quaestiones Mathematicae, we studied using algebraic logic, interpolation, amalgamation using $\alpha$ many variables for topological logic with $\alpha$ many variables briefly $\sf TopL_{\alpha}$. This is a sequel to \cite{3}; the second part on modal cylindric algebras, where we study algebraically other properties of $\sf TopL_{\alpha}$. Modal cylindric algebras are cylindric algebras of infinite dimension expanded with unary modalities inheriting their semantics from a unimodal logic $\sf L$ such as $\sf K5$ or $\sf S4$. Using the methodology of algebraic logic, we study topological (when $\sf L=S4$), in symbols $\sf TCA_{\alpha}$. We study completeness and omitting types $\sf OTT$s for $\sf TopL_{\omega}$ and $\sf TenL_{\omega}$, by proving several representability results for locally finite such algebras. Furthermore, we study the notion of atom-canonicity for both ${\sf TCA}_{n}$ and ${\sf TenL}_n$, a well known persistence property in modal logic, in connection to $\sf OTT$ for ${\sf TopL}_n$ and ${\sf TeLCA}_n$, respectively. We study representability, omitting types, interpolation and complexity isssues (such as undecidability) for topological cylindric algebras. In a sequel to this paper, we introduce temporal cyindric algebras and point out the way how to amalgamate algebras of space (topological algebars) and algebras of time (temporal algebras) forming topological-temporal cylindric algebras that lend themselves to encompassing spacetime gemetries, in a purely algebraic manner.