🤖 AI Summary
This paper resolves the relational representation problem for Sugihara monoids: whether all (including infinitely generated) Sugihara monoids admit faithful representations as algebras of binary relations, with monoidal multiplication interpreted as relational composition. The authors introduce the novel notion of *weakened relations*, constructing a faithful representation of any Sugihara monoid into an algebra of weakened relations, and prove that the monoid operations are exactly realized by relational composition. Combining techniques from order-algebra, FL-algebra theory, ultraproducts, and model theory, they achieve the first uniform relational characterization of the entire class of Sugihara monoids—without restricting to finitely generated instances. Crucially, ultraproduct closure enables lifting finite constructions to the infinite case. The main result establishes that every Sugihara monoid is isomorphic to the reduct of a distributive, involutive FL-algebra over some algebra of weakened relations—thereby fully solving the relational representation problem.
📝 Abstract
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.