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The aim of this paper is to introduce 3-Hom-ρ-Lie algebra structures generalizing the algebras of 3-Hom-Lie algebra. Also, we investigate the representations and deformations theory of this type of Hom-Lie algebras. Moreover, we introduce the definition of extensions and abelian extensions of 3-Hom-ρ-Lie algebras and show that associated to any abelian extension, there is a representation and a 2-cocycle.
Peyghan et al. (Sun,) studied this question.