Representing Semilattices as Relations

Abstract

We represent finite join-semilattices and join-preserving morphisms as a category whose objects and morphisms are binary relations. It is a quotient category of $\mathsf{Rel}_f$'s arrow category, where self-duality arises by taking the relational converse on both objects and morphisms. We investigate the tensor product and the "tight" tensor product using both categories. We also refine the categorical equivalence i.e. finite De Morgan algebras with bounded lattice morphisms are equivalent to a category whose objects are the symmetric relations.