3-nil alternative, pre-Lie, and assosymmetric operads

Alternative algebras are vital for studying and modeling systems that deviate from strict associativity but maintain enough structure to be useful. Indeed, alternative algebras generalize associative algebras by relaxing the strict associativity condition. Alternative algebras naturally include the octonions, which are a key example of a non-associative division algebra. The octonions are part of the Cayley-Dickson construction and play a critical role in geometry, topology, and theoretical physics, especially in string theory and exceptional Lie groups. The origin of alternative algebras lies in the historical exploration of division algebras and their applications extend to various mathematical and physical disciplines, especially in understanding non-associative algebraic structures.In this paper, we consider free alternative algebra with the additional identity x3=0. For motivation, we refer to the dual operad of the alternative operad. Also, we obtain pre-Lie algebra with the identity x3=0 from binary perm algebra. Finally, we consider assosymmetric algebra with identity x3=0.