Commutator equations

Abstract

In this paper, we investigate properties of varieties of algebras described by a novel concept of equation that we call commutator equation. A commutator equation is a relaxation of the standard term equality obtained substituting the equality relation with the commutator relation. Namely, an algebra [Formula: see text] satisfies the commutator equation [Formula: see text] if for each congruence [Formula: see text] in [Formula: see text] and for each substitution [Formula: see text] of elements in the same [Formula: see text]-class, we have [Formula: see text]. This notion of equation draws inspiration from the definition of a weak difference term and allows for further generalization of it. Furthermore, we present an algorithm that establishes a connection between congruence equations valid in the variety generated by the abelian algebras of the idempotent reduct of a given variety and congruence equations that hold in the entire variety. Additionally, we provide a proof that if the variety generated by the abelian algebras of the idempotent reduct of a variety satisfies a nontrivial idempotent Mal’cev condition, then also the entire variety satisfies a nontrivial idempotent Mal’cev condition, a statement that follows also from [ 12 , Theorem 3.13].