 The generalized matrices \(\langle \bA, \cB \rangle\) that correspond to abstract logics have the following two properties: \(A \in \cB\) and \(\cB\) is closed under intersections of arbitrary nonempty families. A family \(\cB\) of subsets of a set \(A\) with these two properties is known as a closed-set system and also as a closure system. There is a dual correspondence between abstract consequence operations on a set \(A\) and closed-set systems on \(A\). Given an abstract consequence operation \(C\) on \(A\), the set \(\cC_C\) of \(C\)-closed sets is a closed-set system and given a closed-set system \(\cC\) the operation \(C_{\cC}\) defined by \(C_{\cC }(X) = \bigcap \{Y \in \cC: X \subseteq Y\}\), for every \(X \subseteq A\), is an abstract consequence operation. In general, every generalized matrix \(\langle \bA, \cB \rangle\) can be turned into a closed-set system by adding to \(\cB \cup \{A\}\) the intersections of arbitrary nonempty subfamilies, and therefore into an abstract logic, which we denote by \(\langle \bA, C_{\cB }\rangle\). In that situation we say that \(\cB\) is a base for \(C_{\cB}\). It is obvious that an abstract logic can have more than one base. Any family of closed sets with the property that every closed set is an intersection of elements of the family is a base. The study of bases for the closed set system of the theories of a logic usually plays an important role in its study. For example, in classical logic an important base for the family of its theories is the family of maximal consistent theories and in intuitionistic logic the family of prime theories. In a similar way, the systematic study of bases for generalized matrix models of a logic becomes important.