Elementary Properties of Free Lattices

TL;DR

It is proved that the free lattices of finite rank are not positively indistinguishable, and that for any lattice $\mathbf K$ which satisfies Whitman's condition $(W)$ and which is generated by join prime elements the lattices $\mathBF K$, $\mathrm{DM}(\mathbf F_n$ and $\mathRM{Id}(\Mathbf K)$ all share the same positive universal theory.

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.

Figures (3)

• Figure 1: Lattice $\mathbf A$ obtained by doubling six elements in $\operatorname{FD}_3$
• Figure 2: Schematic of interval decomposition of $\mathbf F_3$.
• Figure 3: The bounded congruence classes of the natural map of $\mathbf F_3$ onto the pentagon $\mathbf N_5$, where we let $w=x(z+xy)$.

Theorems & Definitions (33)

• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Conjecture 1.6
• Theorem 3.1
• proof
• Definition 4.1
• Definition 4.3
• Lemma 4.5
• proof