@article{Guzzo Jr._Labra_2017, title={An equivalence in generalized almost-Jordan algebras}, volume={35}, url={https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/1325}, DOI={10.4067/S0716-09172016000400011}, abstractNote={<p style="font-size: 13.192px; font-family: verdana, arial;" align="justify"><span style="font-family: verdana; font-size: x-small;">In this paper we work with the variety of commutative algebras satisfying the identity β((x<sup>2</sup>y)x — ((yx)x)x) +γ(x<sup>3</sup>y — ((yx)x)x) = 0, where β, γ are scalars.    They are called generalized almost-Jordan</span></p><p style="font-size: 13.192px; font-family: verdana, arial;" align="justify"><span style="font-family: verdana; font-size: x-small;">algebras. We prove that this variety is equivalent to the variety of commutative algebras satisfying (3β + γ)(G<sub>y</sub>(x,z,t) — G<sub>x</sub>(y,z,t)) + (β + 3γ)(J(x,z,t)y — J(y,z,t)x) = 0, for all x,y,z,t ∈ A, where J(x,y,z) = (xy)z+(yz)x+(zx)y and G<sub>x</sub>(y,z,t) = (yz,x,t)+(yt,x,z)+ (zt,x,y). Moreover, we prove that if A is a commutative algebra, then J (x, z, t)y = J (y, z, t)x, for all x, y, z, t ∈ A, if and only if A is a generalized almost-Jordan algebra for β= 1 and γ = —3, that is, A satisfies the identity (x<sup>2</sup>y)x + 2((yx)x)x — 3x<sup>3</sup>y = 0 and we study this identity. We also prove that if A is a commutative algebra, then G<sub>y</sub>(x,z,t) = G<sub>x</sub>(y,z,t), for all x,y,z,t ∈ A, ifand only if A is an almost-Jordan or a Lie Triple algebra.</span></p>}, number={4}, journal={Proyecciones (Antofagasta, On line)}, author={Guzzo Jr., Henrique and Labra, Alicia}, year={2017}, month={Mar.}, pages={505-519} }