What is the spirit of the cylindric paradigm, as opposed to that of the polyadic one?

TL;DR

This paper categorically distinguishes cylindric-like algebras from polyadic-like algebras, revealing differences in their reduct operators, and presents new results on amalgamation, axiomatizability, and algebraic equivalences.

Contribution

It provides a categorical framework differentiating cylindric and polyadic paradigms, including properties of reduct functors and new algebraic results.

Findings

Neat reduct operator lacks right adjoint in cylindric-like algebras.

Neat reduct operator is strongly invertible in polyadic-like algebras.

Established categorical equivalence between relation algebras with quasi-projections and directed cylindric algebras.

Abstract

We give a categorial definition separating cylindric-like algebras from polyadic-like ones. Viewing the neat reduct operator as a functor, we show that it does not have a right adjoint in the former case, but it is strongly invertible in the second case. Several new results on amalgamation, and non finite axiomatizability are presented for both paradigms. A hitherto categorial equivalence is also given between relation algebras with quasi-projections and Nemeti's directed cylindric algebras for any dimension.