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In two earlier papers we derived congruence formats with regard to transitionsystem specifications for weak semantics on the basis of a decomposition methodfor modal formulas. The idea is that a congruence format for a semantics mustensure that the formulas in the modal characterisation of this semantics arealways decomposed into formulas that are again in this modal characterisation.The stability and divergence requirements that are imposed on many of the knownweak semantics have so far been outside the realm of this method. Stabilityrefers to the absence of a $\tau$-transition. We show, using the decompositionmethod, how congruence formats can be relaxed for weak semantics that arestability-respecting. This relaxation for instance brings the priority operatorwithin 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 circumventthis problem by proving that a congruence format for a stability-respectingweak semantics is also a congruence format for its divergence-preservingcounterpart.

Divide and Congruence III: From Decomposition of Modal Formulas to Preservation of Stability and Divergence