On the definition of quasi-Jordan algebra

TL;DR

This paper investigates the algebraic structure of quasi-Jordan algebras, identifying key polynomial identities of degree up to four, including a new associator-derivation identity, to deepen understanding of their algebraic properties.

Contribution

It determines the polynomial identities of degree ≤ 4 satisfied by the quasi-Jordan product, introducing a new associator-derivation identity.

Findings

Identifies right commutativity and right quasi-Jordan identity

Discovers a new associator-derivation identity

Provides a complete set of identities up to degree 4

Abstract

Velasquez and Felipe recently introduced quasi-Jordan algebras based on the product $a \triangleleft b = \tfrac12 ( a \dashv b + b \vdash a )$ in an associative dialgebra with operations $\dashv$ and $\vdash$. We determine the polynomial identities of degree $\le 4$ satisfied by this product. In addition to right commutativity and the right quasi-Jordan identity, we obtain a new associator-derivation identity.