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 of T is called multiplicative if for any i,j,k ∈ I, we have that
for some r ∈ I. We show that if T admits a multiplicative basis, then it decomposes as the orthogonal direct sum
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.