  We investigate a Belinschi-Nica type semigroup for free and Boolean
max-convolutions. We prove that this semigroup at time one connects limit
theorems for freely and Boolean max-infinitely divisible distributions.
Moreover, we also construct a max-analogue of Boolean-classical Bercovici-Pata
bijection, establishing the equivalence of limit theorems for Boolean and
classical max-infinitely divisible distributions.
