Degrees of maximality of Łukasiewicz-like sentential calculi (Q1239151)

In this paper, by a method of the author, the strengthenings of the so-called Łukasiewicz-like sentential calculi \({\L}_n^I\) are investigated which are determined by \(n\)-valued Łukasiewicz matrices (\(n\geq 2\), \(n\) finite) with a set \(I\) of superdesignated logical values such that \(1\in I\), \(0 \notin I\) (the set \(A_n\) of all logical values is \(\{0,1/(n-1),\dots,(n-2)/(n-1),1\}\). The calculi \({\L}_n^I\) for \(I \neq \{1\}\) are not implicative \(({\L}_n^{\{1\}} = {\L}_n)\). Despite of this fact, matrices analogous to S-algebras of Rasiowa are used. Characterizations of the classes \(\mathrm{Matr}(C_n^I)\) of all S-matrices for some fixed \({\L}_n^I\), are given. For any \({\L}_n^I\), a class \(\mathrm{Matr}^R(C_n^I)\) of matrices, analogous to S-algebras, is then constructed. An equational characterization of \(\mathrm{Matr}^R(C_n^I)\) is the basis of the following finiteness theorem: The degree of maximality of any \({\L}_n^I\) is finite. Another theorem states that the degrees of maximality of the calculi \({\L}_n^I\) and \({\L}_n\) are equal.