The aim of this paper is to introduce the notion of a mock-Lie bialgebra which is equivalent to a Manin triple of mock-Lie algebras. The study of a special case called coboundary mock-Lie bialgebra leads to the introduction the mock-Lie Yang-Baxter equation on a mock-Lie algebra which is an analogue of the classical Yang-Baxter equation on a Lie algebra. Note that a skew-symmetric solution of mock-Lie Yang-Baxter equation gives a mock-Lie bialgebra. Finally, the notation of O-operators are studied to construct skew-symmetric solution of mock-Lie Yang-Baxter equation. M. S. C (2020):16W10, 16T10, 16T15, 16T25, 17B38.

In this paper, we study the representation of ternary Jordan algebras which allows us to introduce the notion of coherent ternary Jordan algebras. Then the [Formula: see text]-operators of ternary Jordan algebras are introduced and the solutions of ternary Jordan Yang–Baxter equation are discussed involving [Formula: see text]-operators. Moreover, ternary pre-Jordan algebras are studied as the algebraic structure behind the [Formula: see text]-operators. Finally, the relations among ternary Jordan algebras and ternary pre-Jordan algebras are established and illustrated by examples.

The main purpose of this paper is to introduce the notion of n-L-dendriform algebra which can be seen as a dendrification of n-pre-Lie algebras by means of O-operators. We investigate the representation theory of n-pre-Lie algebras and provide some related constructions. Furthermore, we introduce the notion of phase space of a n-Lie algebra and show that a n-Lie algebra has a phase space if and only if it is sub-adjacent to a n-pre-Lie algebra. Moreover, we present a procedure to construct (n+1)-pre-Lie algebras from n-pre-Lie algebras equipped with a generalized trace function.