Submission date: 5 October 2023

Abstract:

We start a systematic analysis of the first-order model theory of free lattices. We prove several results. Firstly, we prove that the free lattices of finite rank are not positively indistinguishable, as there is a positive ∃ ∀-sentence true in F_3 and false in F_4. Secondly, we show that every model of Th(F_n) admits a canonical homomorphism into the profinite completion H_n of F_n. Thirdly, we show that H_n is isomorphic to the Dedekind-MacNeille completion of F_n, and that H_n is not positively elementary equivalent to F_n, as there is a positive ∀∃-sentence true in H_n and false in F_n. Finally, we show that DM(F_n) is a retract of Id(F_n), and that for any lattice K which satisfies Whitman's condition (W) and which is generated by join prime elements the lattices K, DM(K) and Id(K) all share the same positive universal theory.

Mathematics Subject Classification: 03C05, 03C64, 06B05

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Full text arXiv 2310.03366: pdf, ps.