The definition of the exact meaning of conceptual modeling concepts is considered a relevant issue since it enhances their effective and appropriate use by designers and facilitates the automatic processing of the models where they are included. Three related concepts that permit to improve the definition of an association in UML and which still lack of a formal semantic definition are: association redefinition, association specialization and association subsetting. This paper formalizes their semantics and points out the similarities and differences that exist among them. The formalization we propose is based on the meta-modelling approach and a semantic domain composed of a set of basic UML concepts and OCL expressions, which have a previous formal definition in the literature and which are well-understood.