Chains of Theories and Companionability
A paper written with Özcan Kasal.
Version of June 2, 2014 [12 pages, size A4], edited in response to a referee's report (and accepted, March 2015, for publication in the Proceedings of the American Mathematical Society). The main changes are:
- Most of the proof of Theorem 6 is now the proof of a lemma of Kolchin cited by León Sánchez.
- The old Theorem 7 is now just a Proposition; therefore the old Theorem 8 is now Theorem 7.
Version of March 12, 2013 [submitted May 15, 2013]; 16 pages of size A5. (The ArXiv version, arXiv:1303.6759 [math.LO], is the same, but on A4 paper.)
Abstract: The theory of fields that are equipped with a countably infinite family of commuting derivations is not companionable; but if the axiom is added whereby the characteristic of the fields is zero, then the resulting theory is companionable. Each of these two theories is the union of a chain of companionable theories. In the case of characteristic zero, the model-companions of the theories in the chain form another chain, whose union is therefore the model-companion of the union of the original chain. However, in a signature with predicates, in all finite numbers of arguments, for linear dependence of vectors, the two-sorted theory of vector-spaces with their scalar-fields is companionable, and it is the union of a chain of companionable theories, but the model-companions of the theories in the chain are mutually inconsistent. Finally, the union of a chain of non-companionable theories may be companionable.
Version of February 25, 2013; 11 pages of size A4.