Abstract
The third normal form (3NF) which was introduced by E. F. Codd is an important normal form for relation schemes in the relational database. It is known [6] that a set of minimal keys of a relation scheme is a Sperner system (some times it is called an antichain) and for an arbitrary Sperner system there exists a relation scheme the set of minimal keys of which is exactly this Sperner system, This paper gives new necessary and sufficient conditions for an arbitrary relation scheme is in 3NF and its set of minimal keys is a given Sperner system.
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