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Register a new userNon elementary classes of relation and cylindric algebras
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Authors
Tarek Sayed Ahmed
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For any pair of ordinals $alpha<beta$, $sf CA_alpha$ denotes the class of cylindric algebras of dimension $alpha$, $sf RCA_{alpha}$ denote the class of representable $sf CA_alpha$s and $sf Nr_alpha CA_beta$ ($sf Ra CA_beta)$ denotes the class of $alpha$-neat reducts (relation algebra reducts) of $sf CA_beta$. We show that any class $sf K$ such that $sf RaCA_omega subseteq sf Ksubseteq RaCA_5$, $sf K$ is not elementary, i.e not definable in first order logic. Let $2<n<omega$. It is also shown that any class $sf K$ such that $sf Nr_nCA_omega cap {sf CRCA}_nsubseteq {sf K}subseteq mathbf{S}_csf Nr_nCA_{n+3}$, where $sf CRCA_n$ is the class of completely representable $sf CA_n$s, and $mathbf{S}_c$ denotes the operation of forming complete subalgebras, is proved not to be elementary. Finally, we show that any class $sf K$ such that $mathbf{S}_dsf Ra CA_omega subseteq {sf K}subseteq mathbf{S}_csf RaCA_5$ is not elementary. It remains to be seen whether there exist elementary classes between $sf RaCA_omega$ and $mathbf{S}_dsf RCA_{omega}$. In particular, for $mgeq n+3$, the classes $sf Nr_nCA_m$, $sf CRCA_n$, $mathbf{S}_dsf Nr_nCA_m$, where $mathbf{S}_d$ is the operation of forming dense subalgebras are not first order definable.
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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.
In this paper, we give new proofs of the celebrated Andreka-Resek-Thompson representability results of certain axiomatized cylindric-like algebras. Such representability results provide completeness theorems for variants of first order logic, that can also be viewed as multi-modal logics. The proofs herein are combinatorial and we also use some techniques from game theory.
We study the saturation properties of several classes of $C^*$-algebras. Saturation has been shown by Farah and Hart to unify the proofs of several properties of coronas of $sigma$-unital $C^*$-algebras; we extend their results by showing that some coronas of non-$sigma$-unital $C^*$-algebras are countably degree-$1$ saturated. We then relate saturation of the abelian $C^*$-algebra $C(X)$, where $X$ is $0$-dimensional, to topological properties of $X$, particularly the saturation of $CL(X)$.
Let $2<n<mleq omega$. Let $CA_n$ denote the class of cylindric algebras of dimension $n$ and $RCA_n$ denote the class of representable $CA_n$s. We say that $Ain RCA_n$ is representable up to $m$ if $CmAtA$ has an $m$-square representation. An $m$ square represenation is locally relativized represenation that is classical locally only on so called $m$-squares. Roughly if we zoom in by a movable window to an $m$ square representation, there will become a point determinded and depending on $m$ where we mistake the $m$ square-representation for a genuine classical one. When we zoom out the non-representable part gets more exposed. For $2<n<m<lleq omega$, an $l$ square represenation is $m$-square; the converse however is not true. The variety $RCA_n$ is a limiting case coinciding with $CA_n$s having $omega$-square representations. Let $RCA_n^m$ be the class of algebras representable up to $m$. We show that $RCA_n^{m+1}subsetneq bold RCA_n^m$ for $mgeq n+2$.
When classes of structures are not first-order definable, we might still try to find a nice description. There are two common ways for doing this. One is to expand the language, leading to notions of pseudo-elementary classes, and the other is to allow infinite conjuncts and disjuncts. In this paper we examine the intersection. Namely, we address the question: Which classes of structures are both pseudo-elementary and $mathcal{L}_{omega_1 omega}$-elementary? We find that these are exactly the classes that can be defined by an infinitary formula that has no infinitary disjunctions.
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