The Bruhat order on the Schubert varieties of a flag manifold or a Grassmannian was first studied by Ehresmann (1934), and the analogue for more general semisimple algebraic groups was studied by Chevalley (1958).
The left and right weak Bruhat orderings were studied by Björner (1984).
If (W, S) is a Coxeter system with generators S, then the Bruhat order is a partial order on the group W. The definition of Bruhat order relies on several other definitions: first, reduced word for an element w of W is a minimum-length expression of w as a product of elements of S, and the length ℓ(w) of w is the length of its reduced words.
(One could also define the Bruhat graph using multiplication on the right; as graphs, the resulting objects are isomorphic, but the edge labelings are different.)
The strong Bruhat order on the symmetric group (permutations) has Möbius function given by