Partial mirror symmetry, lattice presentations and algebraic monoids

TL;DR

This paper develops the theory of reflection monoids, providing presentations for these monoids and related structures like face lattices, geometric lattices, and algebraic monoids, advancing algebraic and geometric understanding.

Contribution

It introduces new presentations for reflection monoids, join-semilattices, and Renner monoids, connecting algebraic monoids with geometric lattice structures.

Findings

Presented general presentations for reflection monoids.

Derived presentations for face lattices and intersection lattices.

Provided a new perspective on algebraic monoids via lattice theory.

Abstract

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 arXiv:0812.2789. 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.