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Representing Sugihara monoids via weakening relations

We show that all Sugihara monoids can be represented as algebras of binary relations, with the monoid operation given by relational composition. Moreover, the binary relations are weakening relations. The first step is to obtain an explicit relational representation of all finite odd Sugihara chains. Our construction mimics that of Maddux (2010), where a relational representation of the finite even Sugihara chains is given. We define the class of representable Sugihara monoids as those which can be represented as reducts of distributive involutive FL-algebras of binary relations. We then show that the class of representable distributive involutive FL-algebras is closed under ultraproducts. This fact is used to demonstrate that the two infinite Sugihara monoids that generate the quasivariety are also representable. From this it follows that all Sugihara monoids are representable.