Given an equivalence relation $\rho$, we investigate modelling the negation of $\rho$ through an equivalence relation $\rho'$ so that $\rho\wedge\rho'$ is the identity, and $\rho$ together with $\rho'$ is comprehensive, i.e., $\rho\vee\rho'$ covers the entire space. This looks very much like complementation, but the lattice of equivalence relations is complemented only in very rare special cases, in particular under the constraints we impose. We are interested in congruences for stochastic relations, so we investigate from the starting point above under which conditions a congruence can be negated, ending up with a variety of constructions that all are more or less intended to characterize non-congruent objects.

Since congruences are based on countably generated equivalence relations, and these relations are in a one-to-one correspondence with countably generated $\sigma$-algebras, we briefly study these $\sigma$-algebras first. We can then carry results for these $\sigma$-algebras over to the space of equivalence relations. This is the technical starting point. Techniques developed for factoring stochastic relations based on those investigated in Universal Algebra help then to bring forth the desired results.

The results are applied to morphisms, a simple version von Hennessy-Milner logic serves as an illustration.