This is the second in a series of papers of the authors that develops the theory of reflection monoids, motivated by the theory of reflection groups. Reflection monoids were first introduced in [Adv. Math. 223, No. 5, 1782-1814 (2010; Zbl 1238.20073)], in which the authors initiated the formal study of partial mirror symmetry via the theory of what is referred to as reflection monoids.
The aim of the study is three-fold: (i) to wrap up a reflection group and a naturally associated combinatorial object into a single algebraic entity having nice properties, (ii) to unify various unrelated (until now) parts of the theory of inverse monoids under one umbrella, and (iii) to provide workers interested in partial symmetry with the appropriate tools to study the phenomenon systematically.