  In this paper, we introduce PM-mapping class monoids. Braid groups and
mapping class groups have many features in common. Similarly to the notion of
braid PM-monoid, PM-mapping class monoid is defined. This construction is an
analogy of inverse mapping class monoid defined by R. Karoui and V. V.
Vershinin. As the main result, we give the analog of the Dehn-Nilsen-Baer
theorem.
