On the Properties of Join Dependencies

In this chapter we introduce a formal system for the family of functional and join dependencies. The system contains one axiom and three inference rules. A few additional useful rules are derived. It is shown that various formal systems for subfamilies of the family of join dependencies can be derived from our system. Special attention is paid to the case where the dependencies are total. For this case it is shown that our rules derive the complete system for multivalued dependencies. The concepts of left side and right side of multivalued dependency are generalized to stems and branches of a join dependency. The notion of a dependency basis is also extended. Finally, several completeness results concerning the system are proved, e.g., completeness of a projection rule. The main result is that our system is complete for deriving functional and multivalued dependencies from functional and join dependencies and for deriving branches of join dependencies.

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