Lattice properties of congruences for stochastic relations

We have a look at the set of congruences for a stochastic relation; conditions under which the infimum or the supremum of two congruences is a congruence again are investigated. Congruences are based on smooth equivalence relations, and consequences of the observation that the supremum of two smooth relations may fail to be smooth are discussed: analytic spaces are not closed under pushouts, and the set of countably generated σ-algebras is not closed under finite intersections.