One way is to suppose that the Lie algebra can be given a totally ordered basis, that is, it is the free vector space of a totally ordered set. Recall that a free vector space is defined as the space of all finite supported functions from a set to the field (finitely supported means that only finitely many values are non-zero); it can be given a basis formula_123 such that formula_124 is the indicator function for formula_125. Let formula_126 be the injection into the tensor algebra; this is used to give the tensor algebra a basis as well. This is done by lifting: given some arbitrary sequence of formula_127, one defines the extension of formula_102 to be