Kleisli morphisms and randomized congruences for the Giry monad

Stochastic relations are the Kleisli morphisms for the Giry monad. This paper proposes the study of the associated morphisms and congruences. The relationship between kernels of these morphisms and congruences is studied, and a unique factorization of a morphism through this kernel is shown to exist. This study is based on an investigation into countably generated equivalence relations on the space of all subprobabilities. Operations on these relations are investigated quite closely. This utilizes positive convex structures and indicates cross-connections to Eilenberg–Moore algebras for the Giry monad. Hennessy–Milner logic serves as an illustration for randomized morphisms and congruences.