  Matching dependencies were recently introduced as declarative rules for data
cleaning and entity resolution. Enforcing a matching dependency on a database
instance identifies the values of some attributes for two tuples, provided that
the values of some other attributes are sufficiently similar. Assuming the
existence of matching functions for making two attributes values equal, we
formally introduce the process of cleaning an instance using matching
dependencies, as a chase-like procedure. We show that matching functions
naturally introduce a lattice structure on attribute domains, and a partial
order of semantic domination between instances. Using the latter, we define the
semantics of clean query answering in terms of certain/possible answers as the
greatest lower bound/least upper bound of all possible answers obtained from
the clean instances. We show that clean query answering is intractable in some
cases. Then we study queries that behave monotonically wrt semantic domination
order, and show that we can provide an under/over approximation for clean
answers to monotone queries. Moreover, non-monotone positive queries can be
relaxed into monotone queries.
