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Weak implication on generalized Łukasiewicz algebras of order $$n$$. (English) Zbl 1286.03180
Summary: In [Stud. Log. 69, No. 3, 329–338 (2001; Zbl 0995.03047)], T. Almada and J. Vaz de Carvalho introduced the variety $$\mathcal L^m_n$$, $$m \geq 1$$ of $$m$$-generalized Łukasiewicz algebras of order $$n$$ as a generalization of Łukasiewicz algebras of order $$n$$ and a particular case of Ockham algebras. In this note, bearing in mind the important role that weak implication played in the study of Łukasiewicz algebras of order $$n$$, we introduce an implication operation on $$m$$-generalized Łukasiewicz algebras of order $$n$$. As this operation generalizes the one indicated above, we will call it with the same name. The deductive systems associated with this implication enable us to establish an isomorphism between the congruence lattice of an $$m$$-generalized Łukasiewicz algebras of order $$n \, A$$ and the lattice of all the deductive systems of $$A$$. This result turns out to be quite useful for characterizing the principal congruences on these algebras simplier than the one described in the above mentioned paper.
##### MSC:
 03G20 Logical aspects of Łukasiewicz and Post algebras
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