Symmetrization of mono\"ids as hypergroups

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.