On the equational theory of representable polyadic equality algebras

Among others we will prove that the equational theory of omega dimensional representable polyadic equality algebras (RPEA(omega)'s) is not schema axio matizable. This result is in interesting contrast with the Daigneault-Monk representation theorem, which states that the class of representable polyad ic algebras is finite scheme-axiomatizable (and hence the equational theory of this class is finite schema-axiomatizable, as well). We will also show that the complexity of the equational theory of RPEA(omega) is also extreme ly high in the recursion theoretic sense. Finally, comparing the present ne gative results with the positive results of Ildiko sain and Viktor Gyuris [ 12], the following methodological conclusions will be drawn: The negative p roperties of polyadic (equality) algebras can be removed by switching from what we call the "polyadic algebraic paradigm" to the "cylindric algebraic paradigm".