Adding a minimal key to a BCNF decomposition

Question

If I will add a minimal key to a given BCNF decomposition can it be no more a BCNF decomposition?

Formally, say I have F (group of functional dependencies), schema R and it's BCNF decomposition {R1,R2,R3,R4} and I will add another group R5 to the decomposition which is a minimal key of schema R .

Will the new decomposition {R1,R2,R3,R4,R5} be still a BCNF decomposition of R ?

Answer 1

Every minimal key is in BCNF since projection of F+ on it contains only trivial functional dependencies. As observed, eg, by Koehler, any dependency preserving BCNF decomposition can be made into lossless by adding a table representing a minimal key. The resulting decomposition is still dependency preserving.

Henning Köhler: Finding Faithful Boyce-Codd Normal Form Decompositions. AAIM 2006: 102-113