Positive implicative ideals of BCK-algebras based on neutrosophic sets and falling shadows

Abstract

 Neutrosophy is introduced by F. Smarandache in 1980 which studies the origin, nature, and scope of
neutralities, as well as their interactions with different ideational spectra. Neutrosophy considers a proposition, theory,
event, concept, or entity, ”A” in relation to its opposite, ”Anti-A” and that which is not A, ”Non-A”, and that which
is neither ”A” nor ”Anti-A”, denoted by ”Neut-A”. Neutrosophy is the basis of neutrosophic logic, neutrosophic
probability, neutrosophic set, and neutrosophic statistics. In this article, we apply the notion of neutrosophic set
theory to (positive implicative) ideals in BCK-algebras by using the concept of falling shadows. The notions of a
positive implicative (∈, ∈)-neutrosophic ideal and a positive implicative falling neutrosophic ideal are introduced,
and several properties are investigated. Characterizations of a positive implicative (∈, ∈)-neutrosophic ideal are
considered, and relations between a positive implicative (∈, ∈)-neutrosophic ideal and an (∈, ∈)-neutrosophic ideal
are discussed. Conditions for an (∈, ∈)-neutrosophic ideal to be a positive implicative (∈, ∈)-neutrosophic ideal are
provided, and relations between a positive implicative (∈, ∈)-neutrosophic ideal, a falling neutrosophic ideal and a
positive implicative falling neutrosophic ideal are studied. Conditions for a falling neutrosophic ideal to be positive
implicative are provided.