Classification of noncommutative monoid structures on normal affine surfaces

TL;DR

This paper extends the classification of algebraic monoids to noncommutative structures on normal affine surfaces, revealing that all such monoids are toric and detailing their structures via Demazure roots.

Contribution

It introduces a classification of noncommutative monoid structures on normal affine surfaces, showing they are toric and providing explicit descriptions using Demazure roots.

Findings

All two-dimensional algebraic monoids are toric.

Explicit formulas for monoid structures using Demazure roots.

Descriptions of opposite, quotient monoids, and boundary divisors.

Abstract

In 2021, Dzhunusov and Zaitseva classified two-dimensional normal affine commutative algebraic monoids. In this work, we extend this classification to noncommutative monoid structures on normal affine surfaces. We prove that two-dimensional algebraic monoids are toric. We also show how to find all monoid structures on a normal toric surface. Every such structure is induced by a comultiplication formula involving Demazure roots. We also give descriptions of opposite monoids, quotient monoids, and boundary divisors.