Results on Polyadic Algebras

While every polyadic algebra ($\PA$) of dimension 2 is representable, we show that not every atomic polyadic algebra of dimension two is completely representable; though the class is elementary. Using higly involved constructions of Hirsch and Hodkinson we show that it is not elementary for higher dimensions a result that, to the best of our knowledge, though easily destilled from the literature, was never published. We give a uniform flexible way of constructing weak atom structures that are not strong, and we discuss the possibility of extending such result to infinite dimensions. Finally we show that for any finite $n>1$, there are two $n$ dimensional polyadic atom structures $\At_1$ and $\At_2$ that are $L_{\infty,\omega}$ equivalent, and there exist atomic $\A,\B\in \PA_n$, such that $\At\A=\At_1$ and $\At\B= \At_2$, $\A\in \Nr_n\PA_{\omega}$ and $\B\notin \Nr_n\PA_{n+1}$. This can also be done for infinite dimensions (but we omit the proof)