An alternative axiomatic presentation of Nelson algebras

Abstract

Nelson algebras are defined in [Diana Brignole and António Monteiro. Caractérisation des algèbres de Nelson par des égalités. {I}, {II}. Proc. Japan Acad., 43:279--283; 284--285, 1967.] in terms of the language $⟨\wedge ,\vee ,\to ,\sim ,1⟩$. In 1962, Diana Brignole [Diana Brignole. Axiomatización de un $N$-reticulado. Revista de la Unión Matemática Argentina, XXI(3):147--148, 1965.], solved the problem proposed by Antonio Monteiro, of giving an axiomatization of Nelson algebras in terms of the connectives $\rightarrowtail$, $\wedge$ and the constant $0=\sim 1$. In this work, we present for the first time a complete proof of this fact, and also show the dependence and independence of some of the axioms proposed by Brignole.

28 pages