An alternative axiomatic presentation of Nelson algebras

Abstract

Nelson algebras were defined in 1967 by D. Brignole and A. Monteiro in terms of the language ⟨∧,∨,→,∼,1⟩. In 1962, D. Brignole solved the problem, proposed by A. Monteiro, of giving an axiomatization of Nelson algebras in terms of the connectives ↣, ∧ and the constant 0 = ∼1, where the operation ↣ is defined by x ↣ y = (x → y)∧ (∼y → ∼x). 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.