Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Non elementary classes of relation and cylindric algebras   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 $\sf
RaCA_\omega \subseteq \sf K\subseteq RaCA_5$, $\sf K$ 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 $\sf Nr_nCA_\omega \cap {\sf CRCA}_n\subseteq {\sf
K}\subseteq \mathbf{S}_c\sf Nr_nCA_{n+3}$, 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 $\mathbf{S}_d\sf Ra CA_\omega \subseteq
{\sf K}\subseteq \mathbf{S}_c\sf RaCA_5$ is not elementary. It remains to be
seen whether there exist elementary classes between $\sf RaCA_\omega$ and
$\mathbf{S}_d\sf RCA_{\omega}$. In particular, for $m\geq n+3$, the classes
$\sf Nr_nCA_m$, $\sf CRCA_n$, $\mathbf{S}_d\sf Nr_nCA_m$, where $\mathbf{S}_d$
is the operation of forming dense subalgebras are not first order definable.

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The categories of this paper are: math-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: funct-an
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The categories of this paper are: hep-lat
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The categories of this paper are: q-fin
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The categories of this paper are: econ
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The categories of this paper are: gr-qc
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The categories of this paper are: stat
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-bio
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The categories of this paper are: alg-geom
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The categories of this paper are: chem-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: nlin
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The categories of this paper are: eess
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The categories of this paper are: nucl-ex