  These results are a contribution to the model theory of matrix consequence.
We give a semantic characterization of uniform and couniform consequence
relations. These properties have never been treated individually, at least in a
semantic manner. We consider these notions from a purely semantic point of view
and separately, introducing the notion of a uniform bundle/atlas and that of a
couniform class of logical matrices. Then, we show that any uniform bundle
defines a uniform consequence; and if a structural consequence is uniform, then
its Lindenbaum atlas is uniform. Thus, any structural consequence is uniform
if, and only if, it is determined by a uniform bundle/atlas. On the other hand,
any couniform set of matrices defines a couniform structural consequence. Also,
the Lindenbaum atlas of a couniform structural consequence is couniform. Thus,
any structural consequence is couniform if, and only if, it is determined by a
couniform bundle/atlas. We then apply these observations to compare structural
consequence relations that are defined in different languages when one language
is a primitive extension of another. We obtain that for any structural
consequence defined in a language having (at least) a denumerable set of
sentential variables, if this consequence is uniform and couniform, then it and
the \emph{ W\'{o}jcicki's consequence} corresponding to it, which is defined in
any primitive extension of the given language, are determined by one and the
same atlas which is both uniform and couniform.
