  We study a certain monoid of endofunctors of the stable homotopy category
that includes localizations with respect to finite unions of Morava
$K$-theories. We work in an axiomatic framework that can also be applied to
analogous questions in equivariant stable homotopy theory. Our results should
be helpful for the study of transchromatic phenomena, including the Chromatic
Splitting Conjecture. The combinatorial parts of this work have been formalised
in the Lean proof assistant.
