  In this paper, we generalize modal $\mu$-calculus to the non-distributive
(lattice-based) modal $\mu$-calculus and formalize some scenarios regarding
categorization using it. We also provide a game semantics for the developed
logic. The proof of adequacy of this game semantics proceeds by generalizing
the unfolding games on the power-set algebras to the arbitrary lattices and
showing that these games can be used to determine the least and the greatest
fixed points of a monotone operator on a lattice. Finally, we define a notion
of bisimulations on the polarities and show invariance of non-distributive
modal $\mu$-calculus under them.
