Completely representable neat reducts

Abstract

For an ordinal $\alpha$, $\sf PEA_{\alpha}$ denotes the class of polyadic equality algebras of dimension $\alpha$. We show that for several classes of algebras that are reducts of $\PEA_{\omega}$ whose signature contains all substitutions and finite cylindrifiers, if $\B$ is in such a class, and $\B$ is atomic, then for all $n<\omega$, $\Nr_n\B$ is completely representable as a $\PEA_n$. Conversely, we show that for any $2<n<\omega$, 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.