On Monoids in the Category of Sets and Relations

The category Rel is the category of sets (objects) and relations (morphisms). Equipped with the direct product of sets, Rel is a monoidal category. Moreover, Rel is a locally posetal 2-category, since every homset Rel(A, B) is a poset with respect to inclusion. We examine the 2-category of monoids RelMon in this category. The morphism we use are lax. This category includes, as subcategories, various interesting classes: hypergroups, partial monoids (which include various types of quantum logics, for example effect algebras) and small categories. We show how the 2-categorical structure gives rise to several previously defined notions in these categories, for example certain types of congruence relations on generalized effect algebras. This explains where these definitions come from.