Quasi-Jordan algebras

In this article we introduce a new algebraic structure of Jordan type and we show several examples. This new structure, called "quasi-Jordan algebras," appears in the study of the product where x,y are elements in a dialgebra (D, (sic), proves). The quasi-Jordan algebras are a generalization of Jordan algebras where the commutative law is changed by a quasi-commutative identity and a special form of the Jordan identity is retained. We show a few results about the relationship between Jordan algebras and quasi-Jordan algebras. Also, we compare quasi-Jordan algebras with some structures. In particular, we find a special relation with Leibniz algebras. We attach a quasi-Jordan algebra to any ad-nilpotent element of index of nilpotence at most 3 in a Leibniz algebra.