Some Characterizations of Finitely Specifiable Implicational Dependency Families

Let r be a relation for the relation scheme R(A1,A2,…,An); then we define Dom(r) to be Domr(A1)×Domr(A2)×…×Domr(An), where Domr(Ai) for each i is the set of all ith coordinates of tuples of r. A relation s is said to be a substructure of the relation r if and only if Dom(s)∩r = s. The following theorems analogous to Tarski-Fraisse-Vaught\u27s characterizations of universal classes are proved: (i) An implicational dependency family (ID-family) F over the relation scheme R is finitely specifiable if and only if there exists a finite number of relations r1,r2,…,rm for R such that r ∈ F if and only if no ri is isomorphic to a substructure of r. (ii) F is finitely specifiable if and only if there exists a natural number k such that r ∈ F whenever F contains all substructures of r with at most k elements. We shall use these characterizations to obtain a new proof for Hull\u27s (1984) characterization of finitely specifiable ID-families