  Higher-order notions of Kreweras complementation have appeared in the
literature in the works of Krawczyk, Speicher, Mastnak, Nica, Arizmendi,
Vargas, and others. While the theory has been developed primarily for specific
applications in free probability, it also possesses an elegant, purely
combinatorial core that is of independent interest. The present article aims at
offering a simple account of various aspects of higher-order Kreweras
complementation on the basis of elementary arithmetic, (co)algebraic,
categorical and simplicial properties of noncrossing partitions. The main idea
is to see noncrossing partitions as providing an interesting noncommutative
analogue of the interplay between the divisibility poset and the multiplicative
monoid of positive integers. Just as the divisibility poset can be regarded as
the decalage of the multiplicative monoid, we exhibit the lattice of
noncrossing partitions as the decalage of a partial monoid structure on
noncrossing partitions encoding higher-order Kreweras complements. While our
results may be considered known, some of the viewpoints can be regarded as
novel, providing an efficient approach both conceptually and computationally.
