  For a symmetrizable Kac-Moody algebra the category of admissible
representations is an analogue of the category of finite dimensional
representations of a semisimple Lie algebra. The monoid associated to this
category and the category of restricted duals by a generalized Tannaka-Krein
reconstruction contains the Kac-Moody group as open dense unit group and has
similar properties as a reductive algebraic monoid. In particular there are
Bruhat and Birkhoff decompositions, the Weyl group replaced by the Weyl monoid,
[M 1].
  We determine the closure relations of the Bruhat and Birkhoff cells, which
give extensions of the Bruhat order from the Weyl group to the Weyl monoid. We
show that the Bruhat and Birkhoff cells are irreducible and principal open in
their closures. We give product decompositions of the Bruhat and Birkhoff
cells. We define extended length functions which are compatible with the
extended Bruhat orders. We show a generalization of some of the Tits axioms for
twinned BN-pairs.
