This paper deals with the extension of Kozen’s μ-calculus with the so-called “existential bisimulation
quantifier”. By using this quantifier one can express the uniform interpolant of any formula of
the μ-calculus. In this work we provide an explicit form for the uniform interpolant of a disjunctive
formula and see that it belongs to the same level of the fixpoint alternation hierarchy of the μ-calculus
than the original formula. We show that this result cannot be generalized to the whole logic, because
the closure of the third level of the hierarchy under the existential bisimulation quantifier is the whole
μ-calculus. However, we prove that the first two levels of the hierarchy are closed. We also provide
the μ-logic extended with the bisimulation quantifier with a complete calculus.
