Necessary conditions for Ternary Algebras

TL;DR

This paper investigates the necessary conditions for the existence of ternary algebras, revealing that additional restrictions, such as linear relations among anti-commutators, are required beyond known cubic identities.

Contribution

It extends the understanding of ternary algebra conditions by exploring identities beyond ternutators, identifying new restrictions involving anti-commutators.

Findings

Ternary algebras satisfy cubic identities as necessary conditions.

Additional restrictions involve linear relations among anti-commutators.

Permitting more general identities imposes further algebraic constraints.

Abstract

Ternary algebras, constructed from ternary commutators, or as we call them, ternutators, defined as the alternating sum of products of three operators, have been shown to satisfy cubic identities as necessary conditions for their existence. Here we examine the situation where we permit identities not solely constructed from ternutators or nested ternutators and we find that in general, these impose additional restrictions; for example, the anti-commutators or commutators of the operators must obey some linear relations among themselves.