Interpreting arithmetics in the ideal lattice of a free vector lattice $\mathcal F_n$

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.