Iterated chromatic localisation

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.