Results on Polyadic Algebras

TL;DR

This paper investigates the properties of polyadic algebras, showing limitations in their representability and elementary classification across dimensions, and constructing specific atom structures with particular logical equivalences.

Contribution

It demonstrates that not all atomic polyadic algebras are completely representable, and provides a uniform method for constructing weak atom structures that are not strong, extending to infinite dimensions.

Findings

Not all atomic polyadic algebras of dimension two are completely representable.

The class of atomic polyadic algebras is elementary in dimension two but not for higher dimensions.

Existence of atom structures that are $L_{,}$ equivalent but differ in representability properties.

Abstract

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)