On properties described by terms in commutator relation

TL;DR

This paper introduces the concept of commutator equations to analyze algebraic varieties, providing new insights into their properties and an algorithm linking congruence equations across related algebraic structures.

Contribution

It defines commutator equations as a relaxation of standard equations, generalizes the weak difference term, and presents an algorithm connecting congruence equations in varieties generated by abelian algebras.

Findings

Commutator equations extend the notion of algebraic equations using the commutator relation.

An algorithm links congruence equations in varieties generated by abelian algebras to those in the entire variety.

Varieties satisfying certain Mal'cev conditions in the abelian algebra subvariety also satisfy them globally.

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}.