In this talk I will present a certain hierarchy of diagrammatic monoids which can be unified by the observation that they all are finite quotients of singular Artin monoids of a certain fixed Coxeter type. In type A, these include a certain submonoid of binary relations, the dual symmetric  inverse monoid, the symmetric inverse monoid and the Brauer monoid. Various properties of these monoids, including presentations, will be discussed.

Based on a joint work with Helena Jonsson, Elin Persson Westin, Shraddha Srivastava, Mateusz Stroinski and Xiaoyu Zhu.