Neat embeddings as adjoint situations

TL;DR

This paper explores the categorical properties of neat reduct operators in algebraic logic, revealing that they lack right adjoints in cylindric algebras but form equivalences in polyadic algebras, highlighting fundamental structural differences.

Contribution

It demonstrates that the neat reduct functor has no right adjoint in cylindric algebras but is an equivalence in polyadic algebras, clarifying their categorical relationships.

Findings

Neat reduct functor lacks right adjoint in cylindric algebras.

In polyadic algebras, the functor is an equivalence.

Highlights structural differences between cylindric and polyadic algebras.

Abstract

We view the neat reduct operator as a functor that lessens dimensions from CA_{\alpha+\omega} to CA_{\alpha} for infinite ordinals \alpha. We show that this functor has no right adjoint. Conversely for polyadic algebras, and several reducts thereof, like Sain's algebras, we show that the analagous functor is an equivalence.