Elementary Properties of Free Lattices

TL;DR

This paper systematically analyzes the first-order model theory of free lattices, revealing differences in positive elementary properties among various completions and establishing relationships between different lattice constructions.

Contribution

It provides new insights into the elementary properties of free lattices, including distinctions between free lattices and their completions, and relationships among various lattice constructions under positive first-order logic.

Findings

Finite rank free lattices are not positively indistinguishable.

Canonical homomorphisms exist into the profinite-bounded completion.

The profinite-bounded completion is isomorphic to the Dedekind-MacNeille completion.

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.