Finite model theory for pseudovarieties and universal algebra: preservation, definability and complexity

TL;DR

This paper investigates the limits of finite model theory in universal algebra, providing examples of finite algebras with unique axiomatisability properties and demonstrating the undecidability of certain definability problems.

Contribution

It introduces new examples of finite algebras with non-finitely axiomatisable varieties and explores the failure of classical theorems at the finite level, advancing understanding in finite model theory and algebra.

Findings

Finite algebras with non-finitely axiomatisable varieties but finitely axiomatisable among finite algebras.

Counterexamples to classical theorems like Łoś-Tarski and Birkhoff's HSP at the finite level.

Undecidability results for first order definability of pseudovarieties.

Abstract

We explore new interactions between finite model theory and classical streams of universal algebra and semigroup theory. A key result is an example of finite algebras whose variety is not finitely axiomatisable in first order logic, but where the class of finite members are finitely axiomatisable amongst finite algebras. These algebras present a negative solution to a first order formulation of the Eilenberg-Sch\"utzenberger problem, and witness the simultaneous failure of the {\L}os-Tarski Theorem, the SP-Preservation Theorem and Birkhoff's HSP-Preservation Theorem at the finite level. The examples also show that a pseudovariety without any finite pseudoequational basis may be finitely axiomatisable in first order logic amongst finite algebras. Other results include the undecidability of deciding first order definability of the pseudovariety of a finite algebra, and a mapping from any fixed finite template constraint satisfaction problem to a first order equivalent variety membership problem.