SZTAKI Közlemények 34. (1986)
Vu Duc Thi: Remarks on dual dependencies
- 115 - §2. THE FAMILIES OF DUAL DEPENDENCIES Definition 2.1. Let D be a d-family, and R be a relation over Q. Then we say that R reprezents D iff D^=D. Definition 2.2. Let D be a d-family over Q, and (A, B)SD. We say that (AfB) is a maximal left side dependency of D if VA'iACA', (A',B)GD - A' =A . Denote by M(D) the set of all maximal left side dependencies of D. We say that A is a maximal left side of D if there is a В so that (A,B)GM(D). Denote G(D) the set of all maximal left sides of D. A family G of subsets of Í2 is called d-semilattice iff G contains 0,Q, and A,BGG imply AfjBGG. In paper C 2 ], the next theorem is proved. Theorem 2.3. Z2l. Let D be a d-family over Q. Then G(D) is a d-semilattice over Q. Conversely, if G is any d-semilattice, then there exist exactly one d-family D so that G(D)=G, where D = { (A,B) : VCGG : A^C - B^C} Definition 2.4. Let I=P (Q), and I is closed under intersection, i.e. A,B6I—АПВ61. Let MGP (Q) . Denote M+ the set {ПМ':M^M}. We say that M generates I iff M+=I. Definition 2.5. Let R= {h^, . . . ,hm) be a relation over Í2. Let 1SL= {aG : hu (a) ^hj(a), l<i<j<m} . We call N^j the nonequality set of R. Denote by N the family of all non-eqiality sets of R. Practically, it is possible that QGN, there are some KL ^ , which are equal to each other. According to the definition of relation. We obtain N. We assume that N= (Ni^. : l<i< j<m] = {A-j^, . . . , Ak : A±^A ^ for i^j ; i,j=l,...,k} We set S=(A,,...,A^}. Based on the non-equality sets of R we give a necessary and sufficient condition for D =D. We can consider the set of nonк.