An alternative axiomatic presentation of Nelson algebras

TL;DR

This paper provides a complete proof of an alternative axiomatic system for Nelson algebras using fewer connectives, clarifying the logical foundations and independence of axioms.

Contribution

It offers the first complete proof of Brignole's axiomatization of Nelson algebras with specific connectives and analyzes the dependence of its axioms.

Findings

Complete proof of Brignole's axiomatization

Analysis of axiom dependence and independence

Simplified axiomatic presentation of Nelson algebras

Abstract

Nelson algebras are defined in [Diana Brignole and Ant\'onio Monteiro. Caract\'erisation des alg\`ebres de Nelson par des \'egalit\'es. {I}, {II}. Proc. Japan Acad., 43:279--283; 284--285, 1967.] in terms of the language $\langle \land,\lor,\to,\sim,1\rangle$. In 1962, Diana Brignole [Diana Brignole. Axiomatizaci\'on de un $N$-reticulado. Revista de la Uni\'on Matem\'atica 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 $\rightarrowtail$, $\land$ 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.