On the definition of nucleus for dialgebras

TL;DR

This paper extends the theory of Malcev dialgebras by defining a generalized alternative di-nucleus and explores their properties, including a conjecture on their speciality and a dialgebra analogue of Kleinfeld's theorem.

Contribution

It introduces the generalized alternative di-nucleus for 0-dialgebras and formulates a conjecture on Malcev dialgebras' speciality, also generalizing Kleinfeld's theorem to dialgebras.

Findings

Definition of the generalized alternative di-nucleus for 0-dialgebras

Formulation of a conjecture on the speciality of Malcev dialgebras

Proof of a dialgebra analogue of Kleinfeld's theorem

Abstract

Malcev dialgebras have been introduced recently by Bremner, Peresi and S\'anchez-Ortega. In the present paper, we continue their study by introducing the notion of the generalized alternative di-nucleus of a 0-dialgebra. A general conjecture about the speciality of Malcev dialgebras in terms of this di-nucleus 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.