  We establish that the bisimulation invariant fragment of MSO over finite
transition systems is expressively equivalent over finite transition systems to
modal mu-calculus, a question that had remained open for several decades. The
proof goes by translating the question to an algebraic framework, and showing
that the languages of regular trees that are recognized by finitary tree
algebras whose sorts zero and one are finite are the regular ones, ie. the ones
expressible in mu-calculus. This corresponds for trees to a weak form of the
key translation of Wilke algebras to omega-semigroup over infinite words, and
was also a missing piece in the algebraic theory of regular languages of
infinite trees since twenty years.
