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In this paper, we first introduce the notion of a compatible 3-Hom-Lie algebra and characterize it as a Maurer-Cartan element by constructing an appropriate bidifferential graded Lie algebra. Then, we develop the cohomology theory for compatible 3-Hom-Lie algebras. Based on this framework, we investigate linear deformations of such algebras. Finally, we study abelian extensions of compatible 3-Hom-Lie algebras and classify their equivalence classes.
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Li et al. (2026) studied this question.
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