  Systems of fixpoint equations over complete lattices, consisting of (mixed)
least and greatest fixpoint equations, allow one to express a number of
verification tasks such as model-checking of various kinds of specification
logics or the check of coinductive behavioural equivalences.
  In this paper we develop a theory of approximation for systems of fixpoint
equations in the style of abstract interpretation: a system over some concrete
domain is abstracted to a system in a suitable abstract domain, with conditions
ensuring that the abstract solution represents a sound/complete
overapproximation of the concrete solution.
  Interestingly, up-to techniques, a classical approach used in coinductive
settings to obtain easier or feasible proofs, can be interpreted as
abstractions in a way that they naturally fit in our framework and extend to
systems of equations.
  Additionally, relying on the approximation theory, we can provide a
characterisation of the solution of systems of fixpoint equations over complete
lattices in terms of a suitable parity game, generalising some recent work that
was restricted to continuous lattices.
  The game view opens the way to the development of on-the-fly algorithms for
characterising the solution of such equation systems.
