n-Algebras admitting a multiplicative basis

Let (A, <., ..., .>) be an n-algebra of arbitrary dimension and over an arbitrary base field F. A basis B = {e(i)}(i is an element of I) of A is said to be multiplicative if for any i(1), ..., i(n) is an element of I, we have either < e(i1), ..., e(in)> = 0 or 0 not equal < e(i1), ..., e(in)> is an element of Fe-j for some (unique) j is an element of I. If n = 2, we are dealing with algebras admitting a multiplicative basis while if n = 3 we are speaking about triple systems with multiplicative bases. We show that if A admits a multiplicative basis then it decomposes as the orthogonal direct sum A = circle plus(alpha) I-alpha, of well-described ideals admitting each one a multiplicative basis. Also, the minimality of A is characterized in terms of the multiplicative basis and it is shown that, under a mild condition, the above direct sum is by means of the family of its minimal ideals.