Completely representable neat reducts

For an ordinal $α$, $\sf PEA_α$ denotes the class of polyadic equality algebras of dimension $α$. We show that for several classes of algebras that are reducts of $\PEA_ω$ whose signature contains all substitutions and finite cylindrifiers, if $\B$ is in such a class, and $\B$ is atomic, then for all $n<ω$, $\Nr_n\B$ is completely representable as a $\PEA_n$. Conversely, we show that for any $2<n<ω$, and any variety $\sf V$, between diagonal free cylindric algebras and quasipolyadic equality algebras of dimension $n$, the class of completely representable algebras in $\sf V$ is not elementary.

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