Weak Implication on Generalized Lukasiewicz Algebras of Order

J. Vaz De Carvalho and T. Almada in A generalization of the Lukasiewicz algebras, Studia Logica 69 (2001), 329-338 introduced the variety Ln , m ≥ 1, n ≥ 2, of m–generalized Lukasiewicz algebras of order n as a generalization of Lukasiewicz 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 Lukasiewicz algebras of order n, we introduce an implication operation on m–generalized Lukasiewicz 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 Lukasiewicz 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.