The class of 2-dimensional neat reducts is not elementary

Abstract

SC, CA, QA and QEA stand for the classes of Pinter's substitution algebras, Tarski's cylindric algebras, Halmos' quasipolyadic algebras and Halmos' quasi- polyadic algebras with equality, respectively. Generalizing a result of Andr eka and N emeti on cylindric algebras, we show that for K2fSC;QA;CA;QEAg and any > 2 the class of 2-dimensional neat reducts of -dimensional algebras in K is not closed under forming elementary subalgebras, hence is not elementary. Whether this result extends to higher dimensions is open.