Some Characterizations of Finitely Specifiable Implicational Dependency Families

Abstract Let r be a relation for the relation scheme R(A 1 ,A 2 ,…,A n ); then we define Dom(r) to be Dom r (A 1 )×Dom r (A 2 )×…×Dom r (A n ), where Dom r (A i ) for each i is the set of all i th 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's 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 r 1 ,r 2 ,…,r m for R such that r ∈ F if and only if no r i 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's (1984) characterization of finitely specifiable ID-families.