Geodesic
Interpreting arithmetics in the ideal lattice of a~free vector lattice~$\mathcal F_n$
Algebra i logika, Tome 47 (2008) no. 1, pp. 71-82.

Voir la notice de l'article provenant de la source Math-Net.Ru

A vector space $V$ over a real field $\mathbf R$ is a lattice under some partial order, which is referred to as a vector lattice if $u+(v\vee w)=(u+v)\vee(u+w)$ and $u+(v\wedge w)=(u+v)\wedge(u+w)$ for all $u,v,w\in V$. It is proved that a model $\mathbf N$ of positive integers with addition and multiplications is relatively elementarily interpreted in the ideal lattice $\mathcal{LF}_n$ of a free vector lattice $\mathcal F_n$ on a set of $n$ generators. This, in view of the fact that an elementary theory for $\mathbf N$ is hereditarily undecidable, implies that an elementary theory for $\mathcal{LF}_n$ is also hereditarily undecidable.
Keywords: vector lattice, free lattice, ideal lattice. 
@article{AL_2008_47_1_a3,
     author = {O. A. Kuryleva},
     title = {Interpreting arithmetics in the ideal lattice of a~free vector lattice~$\mathcal F_n$},
     journal = {Algebra i logika},
     pages = {71--82},
     publisher = {mathdoc},
     volume = {47},
     number = {1},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2008_47_1_a3/}
}
TY  - JOUR
AU  - O. A. Kuryleva
TI  - Interpreting arithmetics in the ideal lattice of a~free vector lattice~$\mathcal F_n$
JO  - Algebra i logika
PY  - 2008
SP  - 71
EP  - 82
VL  - 47
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AL_2008_47_1_a3/
LA  - ru
ID  - AL_2008_47_1_a3
ER  - 
%0 Journal Article
%A O. A. Kuryleva
%T Interpreting arithmetics in the ideal lattice of a~free vector lattice~$\mathcal F_n$
%J Algebra i logika
%D 2008
%P 71-82
%V 47
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AL_2008_47_1_a3/
%G ru
%F AL_2008_47_1_a3
O. A. Kuryleva. Interpreting arithmetics in the ideal lattice of a~free vector lattice~$\mathcal F_n$. Algebra i logika, Tome 47 (2008) no. 1, pp. 71-82. http://geodesic.mathdoc.fr/item/AL_2008_47_1_a3/

[1] G. Birkhoff, “On the structure of abstract algebras”, Proc. Cambridge Philos. Soc., 31 (1935), 433–454 | DOI

[2] A. Grzegorczyk, “Undecidability of some topological theories”, Fundam. Math., 38 (1951), 137–152 | MR

[3] G. Panti, “Prime ideals in free $l$-groups and free vector lattices”, J. Algebra, 219:1 (1999), 173–200 | DOI | MR | Zbl

[4] N. Ya. Medvedev, “Elementarnaya teoriya reshetok $l$-idealov abelevykh $l$-grupp”, Algebra i logika, 44:5 (2005), 540–559 | MR | Zbl

[5] A. Tarski, Undecidable theories, North-Holland Publ. Co., Amsterdam, 1968 | MR | Zbl

[6] K. Baker, “Free vector lattices”, Can. J. Math., 20:1 (1968), 58–66 | MR | Zbl

[7] W. M. Beynon, “Duality theorems for finitely generated vector lattices”, Proc. Lond. Math. Soc. (3), 31:1 (1975), 114–128 | DOI | MR | Zbl

[8] W. M. Beynon, “Application of duality in the theory of finitely generated lattice-ordered Abelian groups”, Can. J. Math., 29 (1977), 243–254 | MR | Zbl

[9] Yu. L. Ershov, Problemy razreshimosti i konstruktivnye modeli, Nauka, M., 1980 | MR