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.

翻译：我们证明所有菅原幺半群均可表示为二元关系代数，其幺半群运算由关系复合给出。此外，这些二元关系均为弱化关系。第一步是获得所有有限奇数菅原链的显式关系表示。我们的构造模仿了Maddux（2010）对有限偶数菅原链给出关系表示的方法。我们将可表示菅原幺半群类定义为那些可作为二元关系的分配对合FL-代数约化子表示的对象。随后证明可表示分配对合FL-代数类在超积运算下封闭。利用这一事实，我们论证了生成拟簇的两个无限菅原幺半群同样具有可表示性。由此可得所有菅原幺半群均具有可表示性。