article · International Journal of Algebra and Computation · 2024

Commutator equations

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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].

Cite this paper

APA 7

Fioravanti, S. (2024). Commutator equations. International Journal of Algebra and Computation, 34(08), 1273–1291. https://doi.org/10.1142/s0218196724500541

MLA 9

Fioravanti, Stefano. “Commutator equations.” International Journal of Algebra and Computation, vol. 34, no. 08, 2024, pp. 1273–91, https://doi.org/10.1142/s0218196724500541.

Chicago

Fioravanti, Stefano. 2024. “Commutator equations.” International Journal of Algebra and Computation 34 (08): 1273–91. https://doi.org/10.1142/s0218196724500541.

Harvard

Fioravanti, S. (2024) ‘Commutator equations’, International Journal of Algebra and Computation, 34(08), pp. 1273–1291. Available at: https://doi.org/10.1142/s0218196724500541.

BibTeX

@article{fioravanti2024commutator,
  title = {{Commutator equations}},
  author = {Fioravanti, Stefano},
  journal = {International Journal of Algebra and Computation},
  volume = {34},
  number = {08},
  pages = {1273--1291},
  year = {2024},
  doi = {10.1142/s0218196724500541}
}

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