Introduction In (Plotkin 1985) a revitalised approach to domain theory was initiated. Roughly, the idea was to eliminate the bottom from the domains and to keep the functions partially defined. Thus replacing Cppo (the category of small cppos ---posets with a least element and closed under lubs of !-chains--- and continuous functions) with pCpo (the category of small cpos ---posets closed under lubs of !-chains--- and partial continuous functions --- see Subsection 3.1). One important point in the reformulation is the recognition of pCpo Research partially supported by Fundaci'on Antorchas and The British Council grant ARG 2281/14/6, and SERC grant RR30735. as a category of partial maps as, for example, such presentation fits better with standard formulations of recursion theory and it allows a categorical description of data types (via partial cartesian closed categories (Longo and Moggi 1984) with finite coproducts) in the presence of fixed-point operators. Following the main moti

We re-examine the modal µ-calculus in the light of some classical theory of Boolean algebras and recent results on duality theory for a modal logic with fixed points. We propose interpreting formulas into a field of subsets of states instead of the full power set lattice used by Kozen. Under this interpretation we relate image compact modal frames with Scott continuity of the box modality, m-saturated transition systems and descriptive modal frames. Also, it is shown that the class of image compact modal frames satisfies the Hennessy-Milner property. We conclude by showing that for descriptive modal µ-frames the standard interpretation coincides with the one we proposed.