On properties described by terms in commutator relation

2023/02/22 by Stefano Fioravanti, Fioravanti, Stefano
Computer Science · #03C05 #08B05 #08B10 #Advanced Algebra and Logic #FOS: Mathematics #Polynomial and algebraic computation #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2302.11448

openalex publication_date 2023/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 A satisfies the commutator equation p ≈_C q if for each congruence theta in Con(A) and for each substitution p^A, q^A of elements in the same θ-class, then (p^A, q^A) ∈ [θ, θ]. This notion of equation draws inspiration from the definition of 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.

Related

• The relationship between two commutators
• Commutator Theory and Abelian Algebras
• Commutator theory for uniformities
• Abelian congruences and similarity in varieties with a weak difference term
• Characterizing the commutator in varieties with a difference term