  In two earlier papers we derived congruence formats with regard to transition
system specifications for weak semantics on the basis of a decomposition method
for modal formulas. The idea is that a congruence format for a semantics must
ensure that the formulas in the modal characterisation of this semantics are
always decomposed into formulas that are again in this modal characterisation.
The stability and divergence requirements that are imposed on many of the known
weak semantics have so far been outside the realm of this method. Stability
refers to the absence of a $\tau$-transition. We show, using the decomposition
method, how congruence formats can be relaxed for weak semantics that are
stability-respecting. This relaxation for instance brings the priority operator
within the range of the stability-respecting branching bisimulation format.
Divergence, which refers to the presence of an infinite sequence of
$\tau$-transitions, escapes the inductive decomposition method. We circumvent
this problem by proving that a congruence format for a stability-respecting
weak semantics is also a congruence format for its divergence-preserving
counterpart.
