Distributed Description Logics (DDL) enable reasoning with multiple ontologies interconnected by directional semantic mapping, called bridge rules. Bridge rules map concepts of a source ontology into concepts of a target ontology. Concept subsumptions of the source ontology can be propagated according to a propagation pattern expressed by bridge rules into concept subsumptions of the target ontology. In the basic formulation of DDL such a propagation is mostly limited to cases when pairs of

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... logies are directly linked by means of bridge rules. However, when more than two ontologies are involved, one would expect that subsumption propagates along chains of ontologies linked by bridge rules, but the semantics of DDL is too weak to support this behaviour. In a recent study, an adjusted semantics for DDL that supports subsumption propagation through chains of ontologies has been introduced. This study makes use of so called compositional consistency requirement that has been employed before in Package-based description logics. While the results concerning subsumption propagation under the adjusted semantics are encouraging, there are important drawbacks. In this paper we take a wider perspective, and propose a study of several different alternative extensions of the DDL semantics. For each of them we study the formal properties, and we select the one that, according to our analysis, constitutes a good compromise, and for this case we provide a sound and complete tableaux decision procedure.