Space and time via Topological and Tense cylindric algebras

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.