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 sentence true in F3 and false in F4. Secondly, we show that every model of Th(Fn) admits a canonical homomorphism into the profinitebounded completion Hn of Fn. Thirdly, we show that Hn is isomorphic to the Dedekind-MacNeille completion of Fn, and that Hn is not positively elementarily equivalent to Fn, as there is a positive sentence true in Hn and false in Fn. Finally, we show that DM(Fn) is a retract of Id(Fn) and that for any lattice K which satisfies Whitman fs condition (W) and which is generated by join prime elements, the three lattices K, DM(K), and Id(K) all share the same positive universal first-order theory.
Elementary properties of free lattices
Paolini G.
2024-01-01
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 sentence true in F3 and false in F4. Secondly, we show that every model of Th(Fn) admits a canonical homomorphism into the profinitebounded completion Hn of Fn. Thirdly, we show that Hn is isomorphic to the Dedekind-MacNeille completion of Fn, and that Hn is not positively elementarily equivalent to Fn, as there is a positive sentence true in Hn and false in Fn. Finally, we show that DM(Fn) is a retract of Id(Fn) and that for any lattice K which satisfies Whitman fs condition (W) and which is generated by join prime elements, the three lattices K, DM(K), and Id(K) all share the same positive universal first-order theory.
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