Taylor’s Modularity Conjecture and Related Problems for Idempotent Varieties

Abstract

Abstract. We provide a partial result on Taylor's modularity conjecture, and several related problems. Namely, we show that the interpretability join of two idempotent varieties that are not congruence modular is not congruence modular either, and we prove an analogue for idempotent varieties with a cube term. Also, similar results are proved for linear varieties and the properties of congruence modularity, having a cube term, congruence n-permutability for a fixed n, and satisfying a non-trivial congruence identity.

n-Permutability is not join-prime for n ≥ 5

“…In [16] Valeriote and Willard, by supplementing the characterization of a Mal'cev class in [9], proved that the interpretability types of the idempotent n-permutable varieties for n ≥ 2 form also a prime filter in the idempotent case. In [14] Opršal obtained a similar result for idempotent modular varieties. In [10] Kearnes and Szendrei proved that for any n having an n-cube term is a join-prime property in the idempotent case.…”

“…In Problem 1.6 of [14] Opršal asked whether for any given n ≥ 3, n-permutability is join-prime in the lattice of interpretability types of idempotent varieties. For n = 2, the question is settled in the positive by the 2-cube term result of Kearnes and Szendrei in [10].…”

Cube term blockers without finiteness

“…The results in this section were discovered during the 2016 'Algebra and Algorithms' workshop after hearing a talk by Matthew Moore on the joinprimeness among idempotent linear Maltsev conditions of the condition expressing the existence of a cube term. Later, Jakub Opršal pointed us to his preprint [14] where he proves our Corollary 6.3 (among other things). Opršal told us that he learned of our characterization of cube terms in terms of crosses from a talk of Szendrei at the AAA90 conference in Novi Sad in 2015, and then developed a similar characterization of his own which allowed him to prove Corollary 6.3.…”

Hardness results for the subpower membership problem

“…We discuss some consequences of Theorem 3.1. We will prove a characterization of consistent strong linear Maltsev conditions which do not imply the existence of a cube term, similar to the results of Opršal [12] and Moore and McKenzie [11]. We will use this characterization along with Theorem 3.1 to show there exist examples of finite algebras in varieties that are congruence distributive and congruence k-permutable (k ≥ 3) whose SMP is EXPTIME-complete.…”