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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.
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
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 ca
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 c
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 $m$ squ
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 all