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

Authors

Shuker khalil, Department of Mathematics, College of Science, University of Basrah, Basrah 61004, IraqFollow
Ali Abbas Asmae, Department of Mathematics, College of Science, University of Basrah, Basrah 61004, Iraq

Abstract

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 λ_n^β∗𝒜 are defined. Next, we show that if 𝒜 is a permutation ideal, then λ_n^β∗𝒜 is a permutation ideal that contains 𝒜. Also, in any commutative permutation BCK--algebra the inequality (λ_i^β ⊛ λ_j^β) ⊛ λ_m^β = (λ_i^β ⊛ λ_m^β) ⊛ λ_j^β, for any λ_i^β, λ_j^β, λ_m^β ∈ X is provided. After that, we show that (λ_n^β λ_m^β) ⊛ (λ_n^β λ_k^β) ≤ (λ_n^β λ_m^β) ≤ λ_m^β, for any λ_i^β, λ_j^β, λ_m^β ∈ X in any commutative permutation BCK--algebra, where λ_n^β λ_j^β = λ_n^β ⊛ (λ_n^β ⊛ λ_j^β). The notion of triple merge permutation (TMP) in S_{n + k + m} for any permutations β, δ and γ in symmetric groups S_m, S_k and S_n 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.

Recommended Citation

khalil, Shuker and Asmae, Ali Abbas (2025) "The Permutation Annihilator Ideals in Commutative Permutation BCK–Algebras with their Applications," Iraqi Journal for Computer Science and Mathematics: Vol. 6: Iss. 1, Article 6.
DOI: https://doi.org/10.52866/2788-7421.1235
Available at: https://ijcsm.researchcommons.org/ijcsm/vol6/iss1/6