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

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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/}
}
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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/