Elementary Theory for $l$-Ideal Lattices of Abelian $l$-Groups

A description is furnished for ideal lattices of free vector lattices $\mathcal F_n$ with a finite number $n$ of free generators. It is proved that an elementary theory for $l$-ideal lattices of Abelian lattice-ordered groups is undecidable, which gives a negative solution to Problem 5.20 in the “Kourovka Notebook.”