Abstract: 
$SC$, $CA$, $QA$ and $QEA$ stand for the classes of Pinter's substitution algebras, Tarski's cylindric algebras, Halmos' quasipolyadic algebras and Halmos' quasipolyadic algebras with equality, respectively. Generalizing a result of Andr{\'e}ka and N{\'e}meti on cylindric algebras, we show that for $K\in \{SC,QA,CA,QEA\}$ and any $\beta >2$ the class of $2$-dimensional neat reducts of $\beta $-dimensional algebras in $K$ is not closed under forming elementary subalgebras, hence is not elementary. Whether this result extends to higher dimensions is open.

1. Department of Mathematics
   Faculty of Science
   Cairo University
   Giza, Egypt