Symmetrization of mono\"ids as hypergroups

TL;DR

This paper extends the Grothendieck group construction to certain idempotent monoids, resulting in hypergroups that satisfy a universal property and generalize classical constructions.

Contribution

It introduces a novel method to construct hypergroups from specific commutative monoids, including idempotent cases, expanding the algebraic framework.

Findings

Constructs hypergroups from idempotent monoids

Provides a universal property for the hypergroup construction

Aligns with classical Grothendieck group in many cases

Abstract

We adapt the construction of the Grothendieck group associated to a commutative mono\"id to handle idempotent mono\"ids. Our construction works for a restricted class of commutative mono\"ids, it agrees with the Grothendieck group construction in many cases and yields a hypergroup which solves the universal problem for morphisms to hypergroups. It gives the expected non-trivial hypergroup construction in the case of idempotent mono\"ids.