Space and time via Topological and Tense cylindric algebras

TL;DR

This paper explores algebraic logic properties of topological and temporal cylindric algebras, including completeness, representability, and omitting types, with applications to spacetime geometries.

Contribution

It advances the understanding of topological and temporal cylindric algebras, introducing new algebraic properties and methods for combining space and time in a logical framework.

Findings

Proved representability results for locally finite topological cylindric algebras.

Studied atom-canonicity and its connection to omitting types.

Analyzed complexity issues like undecidability in topological 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.