Arbitrary triple systems admitting a multiplicative basis. Issue 3 (4th March 2017) - Calderón, Antonio J.; Navarro, Francisco J.; Sánchez, José M.
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 basisℬ = { e i } i ∈ I of T is called multiplicative if for any i, j, k  ∈  I, we have that⟨ e i, e j, e k ⟩ ∈ 𝔽 e r for some r  ∈  I . We show that if T admits a multiplicative basis, then it decomposes as the orthogonal direct sumT = ⊕ k ℑ 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.