Composition of Different-Type Relations via the Kleisli Category for the Continuation Monad

Abstract

We give the way of composing different types of relational notions under certain condition, for example, ordinary binary relations, up-closed multirelations, ordinary (possibly non-up-closed) multirelations, quantale-valued relations, and probabilistic relations. Our key idea is to represent a relational notion as a generalized predicate transformer based on some truth value in some category and to represent it as a Kleisli arrow for some continuation monad. The way of composing those relational notions is given via identity-on-object faithful functors between different Kleisli categories. We give a necessary and sufficient condition to have such identity-on-object faithful functor.