Quasi-projective relation algebras and directed cylindric algebras are categorially equivalent

TL;DR

This paper establishes a categorical equivalence between quasi-projective relation algebras and directed cylindric algebras, explores their logical and algebraic properties, and demonstrates their superamalgamation property.

Contribution

It introduces a categorical equivalence between two algebraic structures and analyzes their logical implications and algebraic properties.

Findings

Categorical equivalence between quasi-projective relation algebras and directed cylindric algebras

A Godels second incompleteness theorem for finite variable fragments of first-order logic

Directed cylindric algebras reflect differences in set theories like CH and its negation

Abstract

We show that quasi-projective relation algebras and directed cylindric algebras are equivalent categorialy. We work out a Godels second incompleteness theorem for finite varibale fragments of first order logic. We show that distinct set theories (like one with CH, and another with its negation) give rise to equationally distinct simple directed cylindric algebras. This correspondance was worked out for quasi projective relation algebras (with a distinguished element corresponding to membership relation). The idea of the proof is that the Sagi representation of directed cylindric algebras preserve well-foundnes. Finally, using that the functor defined preserves order, we show that the class of directed cylindric algebras have the superamalgmation property.