# Chomp on numerical semigroups

## TL;DR

This paper explores the game of chomp on numerical semigroups, linking game strategies to algebraic properties, and determines winning strategies for various classes of semigroups, including symmetric and arithmetic sequence-generated ones.

## Contribution

It characterizes winning strategies for chomp on different classes of numerical semigroups and proves the decidability of determining the winner for a given semigroup.

## Key findings

- Winning strategies are characterized for symmetric semigroups.
- Winning strategies are characterized for semigroups of maximal embedding dimension.
- Decidability of the winning player for any numerical semigroup is established.

## Abstract

We consider the two-player game chomp on posets associated to numerical semigroups and show that the analysis of strategies for chomp is strongly related to classical properties of semigroups. We characterize, which player has a winning-strategy for symmetric semigroups, semigroups of maximal embedding dimension and several families of numerical semigroups generated by arithmetic sequences. Furthermore, we show that which player wins on a given numerical semigroup is a decidable question. Finally, we extend several of our results to the more general setting of subsemigroups of $\mathbb{N} \times T$, where $T$ is a finite abelian group.