Completely representable neat reducts

Abstract

For an ordinal $\alpha$, $\mathsf{PE}{\mathsf{A}}_{\alpha }$ denotes the class of polyadic equality algebras of dimension $\alpha$. 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 $n<\omega$, $\Nr_n\B$ is completely representable as a $\PEA_n$. Conversely, we show that for any $2<n<\omega$, and any variety $\mathsf{V}$, between diagonal free cylindric algebras and quasipolyadic equality algebras of dimension $n$, the class of completely representable algebras in $\mathsf{V}$ is not elementary.

arXiv admin note: substantial text overlap with arXiv:1504.05947, arXiv:1912.12114, arXiv:1912.12182, arXiv:1408.3282