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Łukasiewicz implication prealgebras. (English) Zbl 1413.03028

Summary: In this paper we revise the Łukasiewicz implication prealgebras which we will call Łukasiewicz \(I\)-prealgebras to sum up. They were used by Antonio Jesús Rodríguez Salas on his doctoral thesis under the name of Sales prealgebras. These structures are a natural generalization of the notion of \(I\)-prealgebras, introduced by A. Monteiro [“Algebras implicativas trivalentes de Łukasiewicz”, Lectures given at the Univ. Nac. del Sur, Bahía Blanca, Argentina, 1968] aiming to study using algebraic techniques the \(\{\rightarrow \}\)-fragment of the three-valued Łukasiewicz propositional calculus. The importance of Łukasiewicz \(I\)-prealgebras focuses on the fact that from these structures we can directly prove that Lindembaun-Tarski algebra in the \(\{\rightarrow \}\)-fragment of the infinite-valued Łukasiewicz implication propositional calculus is a Łukasiewicz residuation BCK-algebra in the sense of J. Berman and W. J. Blok [Stud. Log. 77, No. 2, 153–180 (2004; Zbl 1062.03062)]. This last result is indicated without a proof on Y. Komori’s paper [Rep. Fac. Sci., Shizuoka Univ. 12, 1–5 (1978; Zbl 0377.02021)] and it is suggested on his general lines on the Rodriguez Salas thesis.

MSC:

03G25 Other algebras related to logic
06F35 BCK-algebras, BCI-algebras
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