Relation Liftings in Coalgebraic Modal Logic

In this thesis we study relation liftings in the context of coalgebraic modal logic. In the first part of the thesis we look for conditions on relation liftings that can be used to define a notion of bisimilarity between states in coalgebras, such that two states are bisimilar if and only if they are behaviorally equivalent. We show that this is the case for relation liftings that are lax extensions and additionally preserve diagonal relations. In the second part of the thesis we develop a coalgebraic nabla logic for an arbitrary lax extension. For this logic we prove that, under additional conditions, bisimulation quantifiers are definable in the nabla logic. This has a Uniform Interpolation Theorem as consequence.