Extending the Bruhat order and the length function from the Weyl group to the Weyl monoid

For a symmetrizable Kac–Moody algebra the category of admissible representations of the category is an analogue of the category of finite-dimensional representations of a semisimple Lie algebra. The monoid which is associated to this category and the category of restricted duals by a Tannaka–Krein reconstruction contains the Kac–Moody group as open dense unit group. It has similar properties as a reductive algebraic monoid. In particular, there are Bruhat and Birkhoff decompositions, the Weyl group replaced by the Weyl monoid [C. Mokler, An analogue of a reductive algebraic monoid, whose unit group is a Kac–Moody group, arXiv: math.AG/0204246, Mem. Amer. Math. Soc., in press]. 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 twin BN-pairs.