Elementary properties of free lattices

Abstract

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 𝐅 3 {\mathbf{F}_{3}} and false in 𝐅 4 {\mathbf{F}_{4}} . Secondly, we show that every model of Th ⁢ ( 𝐅 n ) {\mathrm{Th}(\mathbf{F}_{n})} admits a canonical homomorphism into the profinite-bounded completion 𝐇 n {\mathbf{H}_{n}} of 𝐅 n {\mathbf{F}_{n}} . Thirdly, we show that 𝐇 n {\mathbf{H}_{n}} is isomorphic to the Dedekind–MacNeille completion of 𝐅 n {\mathbf{F}_{n}} , and that 𝐇 n {\mathbf{H}_{n}} is not positively elementarily equivalent to 𝐅 n {\mathbf{F}_{n}} , as there is a positive ∀ ∃ {\forall\exists} -sentence true in 𝐇 n {\mathbf{H}_{n}} and false in 𝐅 n {\mathbf{F}_{n}} . Finally, we show that DM ⁢ ( 𝐅 n ) {\mathrm{DM}(\mathbf{F}_{n})} is a retract of Id ⁢ ( 𝐅 n ) {\mathrm{Id}(\mathbf{F}_{n})} and that for any lattice 𝐊 {\mathbf{K}} which satisfies Whitman’s condition ( W ) {\mathrm{(W)}} and which is generated by join prime elements, the three lattices 𝐊 {\mathbf{K}} , DM ⁢ ( 𝐊 ) {\mathrm{DM}(\mathbf{K})} , and Id ⁢ ( 𝐊 ) {\mathrm{Id}(\mathbf{K})} all share the same positive universal first-order theory.