Fragments of the mu calculus
To answer the first part of your question 


  Modal specifications are a syntactic fragment of the Mu-calculus, Guillaume Feuillade, 2005.


From the introductory paragraph


  As a consequence, we define a syntactic fragment of the Mu-calculus, called the conjunctive modal Nu calculus, ...In this report, we prove that the conjunctive modal Nu-calculus and modal specifications have the same expressive power and that we can switch between both without loss of generality. We also prove that the set of models of a modal specification is a lattice with finite models as extrema. 


You should examine the details, particularly because the notion of a modal specification is not standard, and is defined in the article.