Completeness and Definability of a Modal Logic Interpreted over Iterated Strict Partial Orders

Any strict partial order R on a nonempty set X defines a function θR which associates to each strict partial order S ⊆ R on X the strict partial order θR(S) = R ◦ S on X. Owing to the strong relationships between Alexandroff TD derivative operators and strict partial orders, this paper firstly calls forth the links between the CantorBendixson ranks of Alexandroff TD topological spaces and the greatest fixpoints of the θ-like functions defined by strict partial orders. It secondly considers a modal logic with modal operators 2 and 2 respectively interpreted by strict partial orders and the greatest fixpoints of the θ-like functions they define. It thirdly addresses the question of the complete axiomatization of this modal logic.