  In 2010, Everitt and Fountain introduced the concept of reflection monoids.
The Boolean reflection monoids form a family of reflection monoids (symmetric
inverse semigroups are Boolean reflection monoids of type $A$). In this paper,
we give a family of presentations of Boolean reflection monoids and show how
these presentations are compatible with quiver mutations of orientations of
Dynkin diagrams with frozen vertices. Our results recover the presentations of
Boolean reflection monoids given by Everitt and Fountain and the presentations
of symmetric inverse semigroups given by Popova respectively. Surprisingly,
inner by diagram automorphisms of irreducible Weyl groups and Boolean
reflection monoids can be constructed by sequences of mutations preserving the
same underlying diagrams. Besides, we show that semigroup algebras of Boolean
reflection monoids are cellular algebras.
