On the definitions of nucleus for dialgebras

Malcev dialgebras were introduced recently by Bremner, Peresi and Sánchez-Ortega. In the present paper, we continue their study by introducing the notion of the generalized alternative dinucleus of a 0-dialgebra. A general conjecture about the speciality of Malcev dialgebras in terms of this dinucleus is formulated. In the last section, we introduce the appropriate generalization of the associative nucleus for dialgebras, and prove an analogue of Kleinfeldʼs theorem for the setting of dialgebras.