Arbitrary triple systems admitting a multiplicative basis

ABSTRACT

Let (T,⟨⋅,⋅,⋅⟩) be a triple system of arbitrary dimension, over an arbitrary base field 𝔽 and in which any identity on the triple product is not supposed. A basis $\mathcal{ℬ}={\left\{e_i\right\}}_{i\in I}$ of T is called multiplicative if for any i,j,k ∈ I, we have that $⟨e_i,e_j,e_k⟩\in 𝔽e_r$ for some r ∈ I. We show that if T admits a multiplicative basis, then it decomposes as the orthogonal direct sum $T={\oplus }_k{\Im }_k$ of well-described ideals admitting each one a multiplicative basis. Also, the minimality of T is characterized in terms of the multiplicative basis and it is shown that, under a mild condition, the above direct sum is by the family of its minimal ideals.

KEYWORDS:

• Infinite dimensional triple system
• multiplicative basis
• nonassociative triple system
• structure theory
• triple system

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 15A03
• 17A40
• 17A60