The Permutation Annihilator Ideals in Commutative Permutation BCK–Algebras with their Applications

This paper introduces new concepts such as permutation BCK--algebra, permutation involutory ideal, commutative permutation BCK--algebra, and prime permutation ideal. Additionally, their attributes are examined. This paper elucidates a method for determining a relationship between the chemical structure of atoms for the chemical element Cadmium, and some of our suggestions are given here. In this work, the structure of the sets 𝒜 and λnβ∗𝒜 are defined. Next, we show that if 𝒜 is a permutation ideal, then λnβ∗𝒜 is a permutation ideal that contains 𝒜. Also, in any commutative permutation BCK--algebra the inequality (λiβ ⊛ λjβ) ⊛ λmβ = (λiβ ⊛ λmβ) ⊛ λjβ, for any λiβ, λjβ, λmβ ∈ X is provided. After that, we show that (λnβ λmβ) ⊛ (λnβ λkβ) ≤ (λnβ λmβ) ≤ λmβ, for any λiβ, λjβ, λmβ ∈ X in any commutative permutation BCK--algebra, where λnβ λjβ = λnβ ⊛ (λnβ ⊛ λjβ). The notion of triple merge permutation (TMP) in Sn + k + m for any permutations β, δ and γ in symmetric groups Sm, Sk and Sn respectively is shown and then the concepts of triple merge permutation BCK--algebra and triple merge permutation bounded are studied. Also, the atomic shells for three chemical elements they are Carbon, Beryllium and Hydrogen are studied and the structure of the triple merge permutation is given from their atomic shells. Some interesting inequalities in commutative permutation BCK--algebra and in triple merge permutation BCK--algebra are given to support this work.