Elementary properties of free lattices

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>∃</m:mo> <m:mo>∀</m:mo> </m:mrow> </m:math> {∃∀} -sentence true in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝐅</m:mi> <m:mn>3</m:mn> </m:msub> </m:math> {F₃} and false in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝐅</m:mi> <m:mn>4</m:mn> </m:msub> </m:math> {F₄} . Secondly, we show that every model of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Th</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>𝐅</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {Th(Fₙ)} admits a canonical homomorphism into the profinite-bounded completion <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝐇</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> {Hₙ} of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝐅</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> {Fₙ} . Thirdly, we show that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝐇</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> {Hₙ} is isomorphic to the Dedekind–MacNeille completion of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝐅</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> {Fₙ} , and that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝐇</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> {Hₙ} is not positively elementarily equivalent to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝐅</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> {Fₙ} , as there is a positive <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>∀</m:mo> <m:mo>∃</m:mo> </m:mrow> </m:math> {∀∃} -sentence true in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝐇</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> {Hₙ} and false in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝐅</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> {Fₙ} . Finally, we show that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>DM</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>𝐅</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {DM(Fₙ)} is a retract of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Id</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>𝐅</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {Id(Fₙ)} and that for any lattice <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝐊</m:mi> </m:math> {K} which satisfies Whitman’s condition <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">W</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(W)} and which is generated by join prime elements, the three lattices <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝐊</m:mi> </m:math> {K} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>DM</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>𝐊</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {DM(K)} , and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Id</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>𝐊</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {Id(K)} all share the same positive universal first-order theory.