Elementary properties of free lattices III: Undecidability of the full theory

TL;DR

This paper proves that the full first-order theory of infinite free lattices is undecidable for all infinite cardinalities greater than or equal to 3, resolving a long-standing open problem.

Contribution

It establishes the undecidability of the full theory of free lattices for all infinite cardinalities, extending previous results on their universal theory.

Findings

Full theory of free lattices is undecidable for all infinite cardinals ≥ 3

Universal theory of infinite free lattices is decidable (from prior work)

Addresses a long-standing open problem in lattice theory

Abstract

In [6] we proved that the universal theory of infinite free lattices is (algorithmically) decidable, leaving open the problem of decidability of the full theory of an (infinite) free lattice. We solve this problem by proving that, for every cardinal $\kappa \geq 3$, the first-order theory of the free lattice $\mathbf{F}_\kappa$ is undecidable.