Based on a concise but comprehensive overview of some fundamental properties required from component-based frameworks, namely compositionality, incrementality, flattening, modularity and expressiveness, we review three modifications of the semantics of glue operators in the Behaviour–Interaction–Priority (BIP) framework. We provide theoretical results and examples illustrating the degree to which the three semantics meet these requirements. In particular, we show that the most recent semantics, based on the offer predicate is the only one that satisfies all of them. The classical and offer semantics are not comparable: there are systems that can be assembled in the classical semantics, but not in the offer one. We present a strict characterisation of the behaviour hierarchy determining the conditions, under which systems in the classical semantics can be transposed into the offer semantics: directly, with minor modifications, by introducing a new type of synchronisation or cannot be transposed at all. The offer semantics allows us to extend the algebras, which are used to model glue operators in BIP, to encompass priorities. This extension uses the Algebra of Causal Interaction Trees, T(P), as a pivot: existing transformations automatically provide the extensions for the Algebra of Connectors. We then extend the axiomatisation of T (P), since the equivalence induced by the new operational semantics is weaker than that induced by the interaction semantics. This extension leads to canonical normal forms for all structures and to a simplification of the algorithm for the synthesis of connectors from Boolean coordination constraints.