  In this paper, we shall introduce two monoids. One is called a PM-monoid
which contains the symmetric group, the other is called a braid PM-monoid which
contains the braid group. We shall develop the theory of PM-monoids and that of
braid PM-monoids. The PM-monoids is obtained in the context of the
compactification of projective linear group defined by Mutsumi Saito. The
structure of PM-monoids is described in terms of matched pairs. We can define
braid PM-monoid using a presentation for the PM-monoid. As main results, we
show that braid PM-monoids are described by geometric braids and we find a
solution to the word problem for the braid PM-monoids.
