Affiliation:
1. Department of Mathematics , 3949 University of Hawaii , Honolulu , HI 96822 , USA
2. Department of Mathematics “Giuseppe Peano” , 241041 University of Torino , Via Carlo Alberto 10 , Torino 10123 , Italy
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.