Abstract 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 τ-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 τ-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.
 We present a general class of process languages for rooted eager bisimulation preorder and prove its congruence result. Also, we present classes of process languages for the rooted versions of several other weak preorders. The process languages we propose are defined by the Ordered SOS method which combines the traditional SOS approach, based on transition rules, with the ordering on transition rules. This new feature specifies the order of application of rules when deriving transitions of process terms. We support and illustrate our results with examples of process operators and process languages which benefit from the OSOS formulation. Structured Operational Semantics (SOS) is a popular method for defining semantics by means of transition rules. An important feature of SOS rules is negative premises, which are crucial in the definitions of such phenomena as priority mechanisms and time-outs. However, the inclusion of negative premises in SOS rules also introduces doubts as to the preferred meaning of SOS specifications. Orderings on SOS rules were proposed by Phillips and Ulidowski as an alternative to negative premises. Apart from the definition of the semantics of positive GSOS rules with orderings, the meaning of more general types of SOS rules with orderings has not been studied hitherto. This paper presents several candidates for the meaning of general SOS rules with orderings and discusses their conformance to our intuition for such rules. We take two general frameworks (rule formats) for SOS with negative premises and SOS with orderings, and present semantics-preserving translations between them with respect to our preferred notion of semantics. Thanks to our semantics-preserving translation, we take existing congruence meta-results for strong bisimilarity from the setting of SOS with negative premises into the setting of SOS with orderings. We further compare the expressiveness of rule formats for SOS with orderings and SOS with negative premises. The paper contains also many examples that illustrate the benefits of SOS with orderings and the properties of the presented definitions of meaning.