In this talk, we shall introduce two monoids. One is called a $\mathbb{P}M$-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 the $\PM$-monoid and that of the braid $\PM$-monoid. We define the $\PM$-monoid using a compactification of projective linear group defined by Mutsumi Saito. The structure of the $\PM$-monoid is described in terms of matched pairs. We define the braid $\PM$-monoid using a presentation for the $\PM$-monoid. As main results, we show that the braid $\PM$-monoid is described by geometric braids and we find a solution to the word problem for the braid $\PM$-monoid.