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 B={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>€Fe_r for some r € I. We show that if T admits a multiplicative basis, then it decomposes as the orthogonal direct sum T=o_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.