Equations implying congruence n-permutability and semidistributivity

Abstract

Abstract. T. Dent, K. Kearnes andÁ. Szendrei have defined the derivative, Σ , of a set of equations Σ and shown, for idempotent Σ, that Σ implies congruence modularity if Σ is inconsistent (Σ |= x ≈ y). In this paper we investigate other types of derivatives that give similar results for congruence n-permutable for some n, and for congruence semidistributivity.

Idempotent n -permutable varieties

“…Kearnes and Kiss [16, Problem P6] have asked whether this characterization is valid for all idempotent varieties. Freese [6,Theorem 8] has recently given a partial confirmation by verifying the equivalence for idempotent linear varieties. In Section 3, we answer the question of Kearnes and Kiss affirmatively.…”

Taylor’s Modularity Conjecture and Related Problems for Idempotent Varieties

“…Unfortunately, we are not able to establish the version of Theorem 1.4(ii) for the filters from the chain. We would also like to note that the condition of being n-permutable for some n can be also formulated as 'having no nontrivial compatible partial order'; this have been attributed to Hagemann, for a proof see [Fre13].…”

Taylor’s modularity conjecture holds for linear idempotent varieties

“…Dent, Kearnes and Szendrei also suggested in [3] that alternative notions of "derivative" might be developed, which could be applied in a similar fashion to other Maltsev properties, such as n-permutability. Ralph Freese ([4]) did just that, by defining the notion of order derivative: 4]). Let Σ be an idempotent set of equations.…”

“…If V 1 ∨ V 2 satisfies a nontrivial congruence identity, then either V 1 or V 2 satisfies a nontrivial congruence identity. [4] created an ordered version of the derivative introduced in [3] and used it to establish, for the property of being n-permutable for some n, some results that are analogous to the ones established in [3] for modularity. Here we make use of the results in [4] to also establish the following result: Theorem 1.3.…”

“…Ralph Freese [4] created an ordered version of the derivative introduced in [3] and used it to establish, for the property of being n-permutable for some n, some results that are analogous to the ones established in [3] for modularity.…”