  We obtain, for the first time, a modular many-valued semantics for combined
logics, which is built directly from many-valued semantics for the logics being
combined, by means of suitable universal operations over partial
non-deterministic logical matrices. Our constructions preserve
finite-valuedness in the context of multiple-conclusion logics whereas,
unsurprisingly, it may be lost in the context of single-conclusion logics.
Besides illustrating our constructions over a wide range of examples, we also
develop concrete applications of our semantic characterizations, namely
regarding the semantics of strengthening a given many-valued logic with
additional axioms, the study of conditions under which a given logic may be
seen as a combination of simpler syntactically defined fragments whose calculi
can be obtained independently and put together to form a calculus for the whole
logic, and also general conditions for decidability to be preserved by the
combination mechanism.
