The lattice of linear subspaces in a vector space V can be provided with a structure of monoid by considering the subspace generated by the union of two subspaces as the monoid operation. When looking at flags in V on could consider the sum of two flags using the monoid operation on subspaces. I guess it is then possible to provide flags with a structure of monoid (algebraic monoid ?). How is reflected this monoid structure in terms of Schubert varities or Schubert classes or Schubert polynomials etc. ? Thank you!