Ultraproducts of Z, with an application to many-valued logic

Up to categorical equivalence, abelian lattice-ordered groups with strong unit coincide with Chang's MV-algebras - the Lindenbaum algebras of the infinite-valued Łukasiewicz calculus. While the property of being a strong unit is not definable even in first-order logic, MV-algebras form an equational class. On the other hand, the addition operation and the translation invariant lattice order of a lattice-ordered group are more amenable than the truncated addition operation of an MV-algebra. In this paper MV-algebraic and group-theoretical techniques are combined to classify and axiomatize all universal classes generated by an infinite totally ordered MV-algebra A such that the quotient of A by its unique maximal ideal is finite. The number of elements of this quotient, and that of the largest finite subalgebra of A turns out to be a complete classifier. The main tool for our results is given by order preserving embeddings of totally ordered groups G into ultrapowers of the additive group of integers, that also preserve the nondivisibility properties of prescribed elements of G.