On anti-ideals and interior anti-ideals in CI/BE/GE-algebras

Abstract

In this paper, we introduce and study the concepts of anti-ideals, bi-anti-ideals, and interior antiideals in CI–, BE–, and GE–algebras. We show that CI/BE/GE–algebras admit no left anti-ideals, and investigate the fundamental properties of right anti-ideals, including their behavior under unions, intersections, and direct products, as well as the existence of maximal right anti-ideals via Zorn’s Lemma. We prove that every right anti-ideal and every interior anti-ideal is a bi-anti-ideal, while the converses fail in general. The main result establishes that a non-empty subset A of a BE/CI–algebra X is an interior anti-ideal if and only if it is a right anti-ideal of X; we show by an explicit counterexample that this equivalence does not extend to general GE–algebras. Finally, we study the behavior of interior anti-ideals under homomorphisms and isomorphisms.