# Flat semigroups and weighted homogeneous surface singularities

## TL;DR

This paper investigates the conditions under which flat semigroups can be realized as numerical semigroups associated with weighted homogeneous surface singularities, establishing a characterization involving quotients of flat semigroups.

## Contribution

It provides a complete characterization of representable flat semigroups as quotients of flat semigroups, linking algebraic properties to geometric realizations.

## Key findings

- A semigroup is representable if and only if it is a quotient of a flat semigroup.
- Characterization of representability in terms of flat semigroup quotients.
- Connection between algebraic semigroup properties and geometric surface singularities.

## Abstract

We consider numerical semigroups associated with normal weighted homogeneous surface singularities with rational homology sphere links. We say that a semigroup is representable if it can be realized in this way. In this article, we study the representability of flat semigroups and prove that a numerical semigroup is representable if and only if it can be written as a quotient of a flat semigroup.