Some generalized type functional dependencies formalized as equality set on matrices

In this paper we deal with functional (f) dependencies and their generalizations, the dual, strong (d, s) and weak (w) dependencies. We give new axioms instead of Armstrong's and Czédli's and axiomatize the full w-families. Our axioms are based on a characterization of equality-sets of matrices. We demonstrate an essential difference between the weak dependency and the rest. We give an estimation for the number of rows which is needed for some antichain of an n-element set to represent it as the set of candidate keys in a relation. Finally, we give a combinatorial characterization of the generating sets with minimal cardinality of full f-families.