Implicative N-deductive systems and annihilators on pre-semi-Nelson algebras

This paper introduces the concept of implicative N-deductive systems of pre-semi-Nelson algebras and explores some of their properties. We examine the relationship between implicative N-deductive systems and N-deductive systems and identify some characterization of implicative N-deductive systems. Additionally, we establish a connection between semi-Nelson algebras and Boolean algebras by using the above concept. Furthermore, we prove that the quotient semi-Nelson algebra induced by an implicative N-deductive system becomes a Boolean algebra under certain conditions. We define the annihilator of a subset of a pre-semi-Nelson algebra and investigate related properties. We demonstrate that the annihilator of an N-deductive system is its pseudo complement, and the set of all involutive N-deductive systems forms a Boolean algebra.