Extensions of the \(\aleph_ 0\)-valued Łukasiewicz propositional logic (Q1309335)

\textit{A. Rose's} genus [J. Lond. Math. Soc. 27, 92-102 (1952; Zbl 0048.244)] and \textit{Y. Komori's} \(S_ n^ \omega\) [Nagoya Math. J. 84, 119- 133 (1981; Zbl 0482.03007)] play important roles in the study of extensions of the \(\aleph_ 0\)-valued Łukasiewicz propositional logic. Nevertheless connections between Rose and Komori had not been made clear. In this paper the author clarifies the connections. The set of formulas valid in an algebra \(A\) is denoted by \(\text{Taut}(A)\). The set of formulas which are theorems of the logic \(L\) is denoted \(\text{Th}(L)\). \(S_ n\) denotes the well-known \(n\)-valued Łukasiewicz matrix. \({\mathbf{\L}}_{\aleph_ 0}\) denotes the \(\aleph_ 0\)-valued Łukasiewicz propositional logic. Let \(G_ \varphi\) denote the genus of a formula \(\varphi\). The main theorem states that if \(\varphi\) is any formula and \(G_ \varphi= \langle B;C\rangle\), then \([\text{Th}({\mathbf{\L}}_{\aleph_ 0}+\varphi)=\bigcap_{i\in C}\text{Taut}(S_ i)\cap\bigcap_{i\in B}\text{Taut}(S_ i^ \omega)]\).