空间与时间的拓扑和紧柱形代数
Space and time via Topological and Tense cylindric algebras
摘要：
设$α$为任意序数，$2<n<ω$。在被《数学学报》接受出版的\cite{3}中，我们简要地研究了用$α$多变量进行拓扑逻辑的代数逻辑、插值、合并和$α$多变量$\sf TopL_α$。这是一个续集\cite{3} ;第二部分模态柱形代数，在那里我们研究代数其他属性的$\sf TopL_α$。模态柱形代数是无限维的柱形代数，用一元模态扩展，从单峰逻辑$\sf L$（如$\sf K5$或$\sf S4$）继承其语义。使用代数逻辑的方法，我们研究符号$\sf TCA_α$中的拓扑（当$\sf L=S4$时）。研究了$\sf TopL_ω$和$\sf TenL_ω$的完备性和省略型$\sf OTT$ s，证明了局部有限的这类代数的几个表示性结果.此外，我们研究了原子规范性的概念，为${\sf TCA}_{n}$和${\sf TenL}_n$，一个众所周知的持久性属性模态逻辑，在$\sf OTT$的${\sf TopL}_n$和${\sf TeLCA}_n$，分别。研究了拓扑柱形代数的可表示性、省略类型、插值和复杂性问题（如不可判定性）.在本文的续集中，我们介绍了时间cylindric代数，并指出如何合并代数的空间（拓扑代数）和代数的时间（时间代数）形成拓扑-时间柱形代数，使自己包含时空gemetries，在一个纯粹的代数方式。
英文摘要：
Let $α$ be an arbritary ordinal, and $2<n<ω$. In \cite{3} accepted for publication in Quaestiones Mathematicae, we studied using algebraic logic, interpolation, amalgamation using $α$ many variables for topological logic with $α$ many variables briefly $\sf TopL_α$. This is a sequel to \cite{3}; the second part on modal cylindric algebras, where we study algebraically other properties of $\sf TopL_α$. 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 $\sf TCA_α$. We study completeness and omitting types $\sf OTT$s for $\sf TopL_ω$ and $\sf TenL_ω$, 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.
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