SZTAKI Közlemények 34. (1986)

Vu Duc Thi: Remarks on dual dependencies

Dd(Fd) is called the full family of dual (functional) К ГЧ. dependencies. Definition 1.2. Let R be a relation over £2, and Ai£2. A is a key of Rif A í- Q. The key A is a minimal key of R if for any R ' f A' (A'=A) : A -*• £2 implies A'=A. R Denote by K_ the set of all minimal keys of R. It is clear К. that К„ forms a Sperner-system over £2. Definition 1.3. Let К be a Sperner-system over Q. We define the set of antikeys of K, denoted by К \ as follows: K_1 = {A^£2 : (B6K) - (b|a) and (A£C) - (3BGK) (B?C)} . Clearly, К ^ is also a Sperner-system over £2. К ^ is called the K. set of antikeys of relation R. Definition 1.4. Let £2 be a finite set, and denote P(£2) its power set. Let Y - P(£2)xp(£2) . Then we say that Y satisfies the F-axioms, if for all A,B,C,Di£2. (F1> (A,A)6Y; (F2) (A,B)6Y, (B,C)6Y -(A, C ) GY ; <F3> (A,B)SY, A^C, DEB - (C , D ) 6Y ; (f4> (A,B)6Y, (C,D)6Y -(AUC,BUD)6Y satisfies the D-axioms, iffor all A,В, (0Х ) (A,A)6Y; (D2) (A,B)GY,(B, C ) 6Y - (A, C ) 6Y ; <D3> (A,B)6Y, CiA, B£D- (C , D ) 6Y ; (D4) (A,B)6Y, (C,D)GY -(AUC,BUD)GY; (D5) (A,0)€Y— A = 0. Definition 1.5. Let Y = P (£2) *P (£2) . We say that Y is a d_ (f-) family over Q if Y satisfies the D-(F_) axioms.

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