Elementary Properties of Free Lattices

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.