  Infamously, the finite and unrestricted implication problems for the classes
of i) functional and inclusion dependencies together, and ii) embedded
multivalued dependencies alone are each undecidable. Famously, the restriction
of i) to functional and unary inclusion dependencies in combination with the
restriction of ii) to multivalued dependencies yield implication problems that
are still different in the finite and unrestricted case, but each are finitely
axiomatizable and decidable in low-degree polynomial time. An important
embedded tractable fragment of embedded multivalued dependencies are
independence atoms that stipulate independence between two attribute sets. We
establish a series of results for implication problems over subclasses of the
combined class of functional and inclusion dependencies as well as independence
atoms. One of our main results is that both finite and unrestricted implication
problems for the combined class of independence atoms, unary functional and
unary inclusion dependencies are axiomatizable and decidable in low-degree
polynomial time.
