A simple proof of the hereditary undecidability of the theory of lattice-ordered abelian groups

In 1967 Gurevich [3] published a proof that the class of divisible Axchimedean lat t ice-ordered abelian groups such that the lattice of carriers is an atomic Boolean algebra has a hereditarily undecidable first-order theory. (He essentially showed the reduct of this class to lattices has a hereditarily undecidable first-order theory: on p. 49 of his paper change z ~ u + v to z ~ u v v in the definition of/Sxy. In late 1978 the author introduced a construction which is useful for showing that numerous varieties (or quasi-varieties) have hereditarily undecidable theories (see [1]). We will use this to quickly derive the following weakened version of the above result. (For terminology on lat t ice-ordered abelian groups see [2]).