  In this paper, we extend Manin and Schechtman's higher Bruhat orders for the
symmetric group to higher Bruhat orders for non-longest words $w$ in $S_n$. We
prove that the higher Bruhat orders of non-longest words are ranked posets with
unique minimal and maximal elements. As in Manin and Schechtman's original
paper, the $k$-th Bruhat order for $w$ is created out of equivalence classes of
maximal chains in its $(k-1)$-st Bruhat order. We also define the second and
third Bruhat orders for arbitrary realizable k-sets, and prove that the second
Bruhat order has a unique minimal and maximal element. Lastly, we also outline
how this extension may guide future research into developing higher Bruhat
orders for affine type A Weyl groups.
