Ramified inverse and planar monoids

Ramified monoids are a class of monoids introduced by the authors. The main motivation for considering these monoids comes from knot theory. Earlier we have studied the ramified monoids of the symmetric group and of the Brauer monoid, among others. This paper studies the ramified monoid of the inverse symmetric monoid, which plays a notable role in knot theory as well. Here is also introduced the notion of planar ramified monoid. In particular, we give presentations for some planar ramified monoids arising from noncrossing set partitions.