Partial mirror symmetry, lattice presentations and algebraic monoids

This is the second in a series of papers that develops the theory of reflection monoids, motivated by the theory of reflection groups. Reflection monoids were first introduced in Everitt and Fountain [Adv. Math. 223 (2010) 1782–1814]. In this paper, we study their presentations as abstract monoids. Along the way, we also find general presentations for certain join-semilattices (as monoids under join), which we interpret for two special classes of examples: the face lattices of convex polytopes and the geometric lattices, particularly the intersection lattices of hyperplane arrangements. Another spin-off is a general presentation for the Renner monoid of an algebraic monoid, which we illustrate in the special case of the ‘classical’ algebraic monoids.