On the Equational Theory of Representable Polyadic Equality Algebras

Among others we will see that the equational theory of ! dimensional rep-resentable polyadic equality algebras (RPEA ! 's) is not schema axiomatizable. This result is in interesting contrast with the Daigneault-Monk representation theorem, which states that the class of representable polyadic algebras is nite schema-axiomatizable (and hence the equational theory of this class is nite schema-axiomatizable, as well). We will also see that the complexity of the equational theory of RPEA ! is also extremely high in the recursion theoretic sense. Finally, comparing the present negative results with the positive results of Ildikk o Sain and Viktor Gyuris 10], the following methodological conclusions will be drawn: the negative properties of polyadic (equality) algebras can be removed by switching from what we call the \polyadic algebraic paradigm" to the \cylindric algebraic paradigm".