 We already encountered the first generalization in  Section 6  when we showed that every logic has a reduced matrix semantics. It leads to the class of algebras \(\bAlg^*\bL\). That its definition is a generalization of the Lindenbaum-Tarski method comes from the realization that the relation \(\theta(T)\), associated with an \(\bL\)-theory, defined in the different completeness proofs in the literature that use the Lindenbaum-Tarski method is in fact the Leibniz congruence of the matrix \(\langle \bFm_L, T \rangle\) and that therefore the matrix \(\langle \bFm/\theta(T), T/\theta(T) \rangle\) is its reduction. As we mentioned in  Section 6,  for every logic \(\bL\), every \(\bL\)-sound class of matrices \(\bM\) that contains all the matrices \(\langle \bFm/\bOmega_{\bFm_L }(T), T/ \bOmega_{\bFm_L }(T) \rangle\), where \(T\) is a theory of \(\bL\), is a complete reduced matrix semantics for \(\bL\). From this perspective the notion of the Leibniz congruence of a matrix can be taken as a generalization to arbitrary matrices of the idea that comes from the Lindenbaum-Tarski procedure of proving completeness. Following this course of reasoning, the class \(\bAlg^*\bL\) of the algebras of the reduced matrix models of a logic \(\bL\) is a very natural class of algebras to associate with \(\bL\). It is the class