Problems on neat embeddings solved by rainbow constructions and Monk algebras

TL;DR

This paper surveys recent advances in algebraic logic, focusing on cylindric algebras, using rainbow and Monk-like constructions to address problems related to neat embeddings, non-atom-canonicity, and axiomatizability.

Contribution

It introduces new methods using rainbow constructions to solve longstanding problems in algebraic logic, including generalizations of Hodkinson, Hirsch, and Hodkinson's results.

Findings

Rainbow constructions solve problems on neat embeddings.

Non-atom-canonicity shown for many varieties of algebras.

Finite variable axiomatizations are impossible for certain classes.

Abstract

This paper is a survey of recent results and methods in (Tarskian) algebraic logic. We focus on cylindric algebras. Fix 2<n<\omega. Rainbow constructions are used to solve problems on classes consisting of algebras having a neat embedding property substantially generalizing seminal results of Hodkinson as well as Hirsch and Hodkinson on atom-canonicity and complete representations, respectively. For proving non-atom-canonicity of infinitely many varieties approximating the variety of representable algebras of dimension n, so-called blow up and blur constructions are used. Rainbow constructions are compared to constructions using Monk-like algebras and cases where both constructions work are given. When splitting methods fail. rainbow constructions are used to show that diagonal free varieties of representable diagonal free algebras of finite dimension n, do no admit universal axiomatizations containing only finitely many variables. Notions of representability, like complete, weak and strong are lifted from atom structures to atomic algebras and investigated in terms of neat embedding properties. The classical results of Monk and Maddux on non-finite axiomatizability of the classes of representable relation and cylindric algebras of finite dimension n are reproved using also a blow up and blur construction. Applications to n-variable fragments of first order logic are given. The main results of the paper are summarized in tabular form at the end of the paper.