Elementary properties of free lattices
J. B. Nation, Gianluca Paolini Β· Forum Mathematicum Β· 2024
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 β β {\exists\forall} -sentence true in π 3 {\mathbf{F}_{3}} and false in π 4 {\mathbf{F}_{4}} . Secondly, we show that every model of Th β’ ( π n ) {\mathrm{Th}(\mathbf{F}_{n})} admits a canonical homomorphism into the profinite-bounded completion π n {\mathbf{H}_{n}} of π n {\mathbf{F}_{n}} . Thirdly, we show that π n {\mathbf{H}_{n}} is isomorphic to the DedekindβMacNeille completion of π n {\mathbf{F}_{n}} , and that π n {\mathbf{H}_{n}} is not positively elementarily equivalent to π n {\mathbf{F}_{n}} , as there is a positive β β {\forall\exists} -sentence true in π n {\mathbf{H}_{n}} and false in π n {\mathbf{F}_{n}} . Finally, we show that DM β’