Carlos Gallardo, Alicia Ziliani, Department of Mathematics, Universidad Nacional del Sur, Avda. Alem 1253, 8000 Bahía Blanca, Argentina, e-mail: gallardo@criba.edu.ar, aziliani@gmail.com
Abstract: T. Almada and J. Vaz de Carvalho (2001) stated the problem to investigate if these Lukasiewicz 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.
Keywords: Lukasiewicz algebra of order $n$; $m$-generalized Lukasiewicz algebra of order $n$; equationally definable principal congruences; implicative extensional propositional calculus; completeness theorem
Classification (MSC 2010): 03G10, 06D99, 03B60
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