The ${\mathcal L}^m_n$-propositional calculus

T. Almada and J. Vaz de Carvalho (2001) stated the problem to investigate if these Łukasiewicz algebras are algebras of some logic system. In this article an affirmative answer is given and the ${\mathcal L}^{m}_{n}$-propositional calculus, denoted by ${\ell ^{m}_{n}}$, is introduced in terms of the binary connectives $\to $ (implication), $\twoheadrightarrow $ (standard implication), $\wedge $ (conjunction), $\vee $ (disjunction) and the unary ones $f$ (negation) and $D_{i}$, $1\leq i\leq n-1$ (generalized Moisil operators). It is proved that ${\ell ^{m}_{n}}$ belongs to the class of standard systems of implicative extensional propositional calculi. Besides, it is shown that the definitions of $L^{m}_{n}$-algebra and ${\ell ^{m}_{n}}$-algebra are equivalent. Finally, the completeness theorem for ${\ell ^{m}_{n}}$ is obtained.