Embedding tensors on 3-Leibniz algebras and their derived algebraic structures and deformations

In this paper, first we introduce the notions of 3-tri-Leibniz algebras and embedding tensors on 3-Leibniz algebras. We show that an embedding tensor gives rise to a 3-tri-Leibniz algebra. Conversely, a 3-tri-Leibniz algebra gives rise to a 3-Leibniz algebra and a representation such that the quotient map is an embedding tensor. Furthermore, any 3-tri-Leibniz algebra can be embedded into an averaging 3-Leibniz algebra. Next, we introduce the notion of 3-tri-Leibniz dialgebras and demonstrate that homomorphic embedding tensors inherently induce 3-tri-Leibniz dialgebras. Finally, we study the linear deformations of embedding tensors by defining first cohomology.