  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.
