Definition 1

Let w \in W. The length of w (with respect to S), denoted by l_S(w) or simply by l(w), is the smallest integer q \geq 0 such that w is the product of a sequence of q elements of S. A reduced decomposition of w (with respect to S) is any sequence s = (s_1, \ldots, s_q) of elements of S such that w = s_1 \cdots s_q and q = l(w).

Thus 1 is the unique element of length 0 and S consists of the elements of length 1.