  This paper proves the existence of potentials of the first and second kind of
a Frobenius like structure in a frame which encompasses families of
arrangements.
  Surprisingly the proof is based on the study of finite sets of vectors in a
finite-dimensional vector space $V$. Given a natural number $m$ and a finite
set $(v_i)$ of vectors we give a necessary and sufficient condition to find in
the set $(v_i)$ $m$ bases of $V$. If $m$ bases in $(v_i)$ can be selected, we
define elementary transformations of such a selection and show that any two
selections are connected by a sequence of elementary transformations.
