On properties described by terms in commutator relation

Abstract

We investigate properties of varieties of algebras described by a novel concept of equation that we call \emph{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 $\mathbf{A}$ satisfies the commutator equation $p \approx_{C} q$ if for each congruence theta in Con(\mathbf{A}) and for each substitution $p^{\mathbf{A}}, q^{\mathbf{A}}$ of elements in the same $\theta$-class, then $(p^{\mathbf{A}}, q^{\mathbf{A}}) \in [\theta, \theta]$. This notion of equation draws inspiration from the definition of \emph{weak difference term} and allows for further generalization of it. Furthermore, we present an algorithm that establishes a connection between congruence equations valid within the variety generated by the abelian algebras of the idempotent reduct of a given variety and congruence equations that hold within 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 non-trivial idempotent Mal'cev condition then also the entire variety satisfies a non-trivial idempotent Mal'cev condition, statement that follows also form \cite[Theorem 3.13]{KK.TSOC}.