Cylindric and polyadic algebras, new perspectives

TL;DR

This paper extends the concept of Monk's schema to infinite dimensions, enabling the transfer of finite-dimensional results to infinite-dimensional cases and introducing new algebraic properties and definitions.

Contribution

It generalizes Monk's schema for infinite dimensions, allowing deep finite-dimensional results to be applied in the infinite case, and provides new algebraic properties and corrections in the field.

Findings

Solved problem 2.12 in Henkin Monk and Tarski for infinite dimensions

Proved algebraic properties like amalgamation for various algebraic systems

Identified and corrected errors in previous publications

Abstract

We generalize the notion of Monk's schema in such a way to integrate finite dimensions. This allows us to lift a plathora of deep results proved for finite dimensions to the infinite dimensional case, like the solution to problem 2.12 in Henkin Monk and Tarski part one, solved by Hirsch and Hodkinson. This lifting argument was already used in a joint paper with Robin Hirsch, but in a narrower context, accepted for publication in the Journal of Symbolic Logic. We also give a general new definition of a schema for infinite dimensions covering Monk's schema and Halmos' schema. Several algebraic properties (like amalgamation) are proved for instances of systems of varieties definable by such a schema like MV algebras, reducts of Heyting polyadic algebras and Ferenczi's cylindric polyadic algebras. Finally, two serious errors in two publications in prestigeous journals are pointed out. One is fixed, the alledged result in the second is weakened.