Arbitrary triple systems admitting a multiplicative basis. Issue 3 (4th March 2017)

• Record Type: Journal Article
• Authors: Calderón, Antonio J.; Navarro, Francisco J.; Sánchez, José M.
• Is Part Of: Communications in algebra. Volume 45:Issue 3(2017)
• Page Start: 1203
• Page End: 1210
• Publication Date: 2017-03-04
• Subjects: Infinite dimensional triple system -- multiplicative basis -- nonassociative triple system -- structure theory -- triple system; 15A03 -- 17A40 -- 17A60; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2016.1175589
• Languages: English
• ISSNs: 0092-7872

Abstract

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.