Elementary Properties of Free Lattices

Abstract

We start a systematic analysis of the first-order model theory of free lattices. Firstly, we prove that the free lattices of finite rank are not positively indistinguishable, as there is a positive $\exists \forall$-sentence true in ${\mathbf{F}}_3$ and false in ${\mathbf{F}}_4$. Secondly, we show that every model of $\mathrm{Th}({\mathbf{F}}_n)$ admits a canonical homomorphism into the profinite-bounded completion ${\mathbf{H}}_n$ of ${\mathbf{F}}_n$. Thirdly, we show that ${\mathbf{H}}_n$ is isomorphic to the Dedekind-MacNeille completion of ${\mathbf{F}}_n$, and that ${\mathbf{H}}_n$ is not positively elementarily equivalent to ${\mathbf{F}}_n$, as there is a positive $\forall \exists$-sentence true in ${\mathbf{H}}_n$ and false in ${\mathbf{F}}_n$. Finally, we show that $\mathrm{DM}({\mathbf{F}}_n)$ is a retract of $\mathrm{Id}({\mathbf{F}}_n)$ and that for any lattice $\mathbf{K}$ which satisfies Whitman's condition $(\mathrm{W})$ and which is generated by join prime elements, the three lattices $\mathbf{K}$, $\mathrm{DM}(\mathbf{K})$, and $\mathrm{Id}(\mathbf{K})$ all share the same positive universal first-order theory.

13 pages