The Algebra of Partial Equivalence Relations

Recent work by the author with Bonchi and Sobociński shows how PROPs of linear relations (subspaces) can be presented by generators and equations via a “cube construction”, based on letting very simple structures interact according to PROP operations of sum, fibered sum and composition via a distributive law. This paper shows how the same construction can be used in a cartesian setting to obtain presentations by generators and equations for the PROP of equivalence relations and of partial equivalence relations.