Idempotent n‐permutable varieties

One of the important classes of varieties identified in tame congruence theory is the class of varieties which are n‐permutable for some n. In this paper, we prove two results: (1) for every n>1, there is a polynomial‐time algorithm that, given a finite idempotent algebra A in a finite language, determines whether the variety generated by A is n‐permutable and (2) a variety is n‐permutable for some n if and only if it interprets an idempotent variety that is not interpretable in the variety of distributive lattices.