# Quantum Semigroups from Synchronous Games

## TL;DR

This paper explores how C*-algebras from synchronous games can form quantum semigroups and groups, revealing new algebraic structures related to quantum families of maps and graph endomorphism games.

## Contribution

It introduces a framework linking synchronous game C*-algebras to quantum semigroups and groups, expanding the understanding of quantum symmetries in combinatorial contexts.

## Key findings

- Quantum families of maps form quantum semigroups in certain cases.
- Under state preservation, these structures become compact quantum groups.
- Application to graph endomorphism games illustrates the theory.

## Abstract

We show that the C*-algebras associated with synchronous games give rise to certain quantum families of maps between the input and output sets of the game. In particular situations (e.g. for graph endomorphism games) these quantum families have a natural quantum semigroup structure and if the condition of preservation of a natural state is added, they are in fact compact quantum groups.