  We study an extension of modal $\mu$-calculus to sets with atoms and we study
its basic properties. Model checking is decidable on orbit-finite structures,
and a correspondence to parity games holds. On the other hand, satisfiability
becomes undecidable. We also show expressive limitations of atom-enriched
$\mu$-calculi, and explain how their expressive power depends on the structure
of atoms used, and on the choice between basic or vectorial syntax.

Logic in Computer Science