Bruhat order of tournaments

We extend the Bruhat order on the set SnSn of permutations (permutation matrices) of {1,2,…,n}{1,2,…,n} and its generalization to classes A(R,S)A(R,S) of (0,1)(0,1)-matrices with row sum vector R and column sum vector S  , to a Bruhat order on classes T(R)T(R) of tournaments with score vector R  . As in the case of the Bruhat order on A(R,S)A(R,S), there are two possible Bruhat orders where one is a refinement of the other. We characterize the cover relation for one of these orders. For a special family of score vectors, we show these Bruhat orders are isomorphic to the partially ordered set of all subsets of a set.