Lukasiewicz Implication Prealgebras

Abstract

In this paper we revise the Lukasiewicz implication prealgebras which we will
call Lukasiewicz I−prealgebras to sum up. They were used by Antonio Jes´us 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 in 1968 aiming to study using algebraic techniques the {!}-fragment of the three-valued Lukasiewicz propositional calculus. The importance of Lukasiewicz I−prealgebras focuses on the fact that from these structures we can directly prove that Lindembaun-Tarski algebra in the {!}-
fragment of the infinite-valued Lukasiewicz implication propositional calculus is a Lukasiewicz residuation BCK-algebra in the sense of Berman and Blok [1]. This last result is indicated without a proof on Komori’s paper ([8]) and it is suggested on his general lines on the Rodriguez Salas thesis.