Idempotent n -permutable varieties

Abstract

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 npermutable for some n if and only if it interprets an idempotent variety that is not interpretable in the variety of distributive lattices.

Deciding Some Maltsev Conditions in Finite idempotent algebras

“…While not all Maltsev conditions are of this form, path Maltsev conditions include many important conditions such as the Maltsev term, ternary majority, Jónsson terms or Gumm terms. While the efficient decidability of having the Maltsev or majority term was known before [9] as was the efficient decidability of a chain of n Hagemann-Mitschke terms [21], our framework of path Maltsev conditions unifies these earlier results and shows that it is similarly tractable to decide if an algebra has a fixed length chain of (classical or directed) Jónsson or Gumm terms. See Corollary 8 for a summary of the Maltsev conditions we can work with.…”

Polynomial-time tests for difference terms in idempotent varieties

“…In [14], Ross Willard and the third author define a "local Hagemann-Mitschke sequence" which they use as the basis of an efficient algorithm for deciding for a given n whether an idempotent variety is n-permutable. In [6], Jonah Horowitz introduced similar local methods for deciding when a given variety satisfies certain Maltsev conditions.…”

“…This is largely due to the efforts of researchers who, over the last three decades, have found ingenious ways to coax computers into solving challenging abstract algebraic decision problems, and to do so very quickly. To give a couple of examples related to our own work, it is proved in [14] (respectively, [4]) that deciding whether a finite idempotent algebra generates a variety that is congruence-n-permutable (respectively, congruence-modular) is tractable. b The present paper continues this effort by presenting an efficient algorithm for deciding whether a finitely generated idempotent variety has a difference term.…”

Taylor’s Modularity Conjecture and Related Problems for Idempotent Varieties

“…Theorem 5.6 ( [VW14]). An idempotent variety is not n-permutable for any n if and only if it is interpretable in the variety of distributive lattices.…”