On the complexity of axiomatizations of the class of representable quasi-polyadic equality algebras

Abstract

Using games, as introduced by Hirsch and Hodkinson in algebraic logic, we give a recursive axiomatization of the class RQPEAα of representable quasi-polyadic equality algebras of any dimension α. Following Sain and Thompson in modifying Andréka’s methods of splitting, to adapt the quasi-polyadic equality case, we show that if Σ is a set of equations axiomatizing RPEAn for \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$2< n <\omega$\end{document} and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$l< n,$\end{document} \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$k < n$\end{document}, k′ < ω are natural numbers, then Σ contains infinitely equations in which − occurs, one of + or · occurs, a diagonal or a permutation with index l occurs, more than k cylindrifications and more than k′ variables occur. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim