Goldblatt-Thomason Theorem for Coalgebraic Graded Modal Logic

Graded modal logic (GML) was originally presented by Kit Fine (1972) to make the modal analogue to counting quantifiers explicit. A graded modal formula ♦k is true at a state w in a Kripke model if there are at least k successor states of w where φ is true. One open problem in GML is to show a Goldblatt-Thomason theorem for it. See M. De Rijke’s notes (2000). Recently, Katsuhiko Sano and Minghui Ma (2010) give a model theoretic proof for a Goldblatt-Thomason theorem for graded modal language under Kripke semantics, i.e., a first-order definable class of Kripke frames is definable by a set of GML formulas iff it is closed under disjoint unions, generated subframes, graded bounded morphic images and graded ultrafilter images. However, in this paper, we show another natural Goldblatt-Thoamson theorem for GML under coalgebraic semantics. The coalgebraic perspective on modal logics has been developed in recent years. Coalgebras are general abstract mathematical models for state-based dynamic systems. It is easy to see that Kripke frames are just powerset coalgebras. For the relationship between coalgebra and modal logic, see Yde Venema (2007) and D. Pattinson (2003). For graded modal logic, multi-graphs as coalgebras are suggested in several papers such as G. D’Agostino and A. Visser (2002), L. Schröder (2007), and L. Schröder and D. Pattinson (2008) etc..