On the definitions of nucleus for dialgebras

Abstract

Communicated by Alberto ElduqueIn memory of Jean-Louis Loday MSC: primary 17D10 secondary 17A20, 17A30, 17A32, 17D05 Keywords:Nonassociative algebras and dialgebras Nucleus Ternary derivations Computer algebra 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 Kleinfeld's theorem for the setting of dialgebras.