The class of 2-dimensional neat reducts is not elementary

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 Andreka and Nemeti on cylindric algebras, we show that for K E {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.