Partial mirror symmetry II: Generators and relations Brent Everitt and John Fountain ⋆ Abstract. We continue our development of the theory of reflection monoids by first deriving a presentation for a general reflection monoid from a result of Easdown, East and Fitzgerald for factorizable inverse monoids. We then derive “Popova” style presentations for reflection monoids built from Boolean hyperplane arrangements and reflection arrangements. Introduction In [3] we initiated the formal study of “partial mirror symmetry”–the theory of monoids gen- erated by partial reflections. The principle acheivements of the theory to date, after identifying and formulating the notion itself, are to observe a number of examples of reflection monoids occuring in nature and determine their orders. In this paper we continue the programme with a general presentation for reflection monoids, which we then interpret for a number of the key examples. Historically, this goes back to Popova [7], who gave a simple presentation for the symmetric inverse monoid I n with generators the transpositions (i,i + 1) ∈ S n (the standard Coxeter generators for S n as a Weyl group) and a single idempotent. Just as the symmetric group is the “simplest” family of finite (real) reflection groups, so the symmetric inverse monoid is the simplest family of finite real reflection monoids. In our language, I n is the Boolean reflection monoid of type A n−1 , just as S n is the Weyl group of type A n−1 . We thus recover Popova’s presentation from our general one, as well as a number of others of course. There are other interesting “geometric” interpretations of the Popova presentation: it was recovered in [2] from a presentation for the “braid monoid” on n strands, much as one recovers the Coxeter presentation for S n from a presentation for Artin’s braid group. This paper is organized as follows: we remember reflection monoid terminology from the first paper in the series in §1. The idempotents in our monoids offer many of the difficulties in writing presentations, so they deserve a special section (§2) of their own. Our general presentation is then Theorem 1 of §3, obtained by massaging a presentation for factorizable inverse monoids obtained recently in [1]. The last two sections, §§4-5 interpret the various ingredients of Theorem 1 and perform a few more simplifications for the Boolean and reflection arrangement monoids. 1. Preliminaries on reflection monoids We summarize the notation and conventions of the first paper in the series: V is a vector space over a field F and W ⊂ GL(V ) a group generated by reflections. The main theorems of the paper in §3 work at this level of generality, but later we will restrict to the case F = R and W finite, in which case W = W (Φ) is determined by a root system Φ in V . In particular we shall Department of Mathematics, University of York, York YO10 5DD, United Kingdom. e-mail: bje1@york.ac.uk (Brent Everitt), e-mail: jbf1@york.ac.uk (John Fountain). ⋆ Some of the results of this paper were obtained while the first author was visiting the Institute for Geometry and its Applications, University of Adelaide, Australia. He is grateful for their hospitality. A grant from the Royal Society made it possible for the second author to visit the University of Adelaide to continue the work reported here. He would like to express his gratitude to the members of the Glenelg Mathematics Institute for their kindness and hospitality during his visit to Adelaide.