id author title date pages extension mime words sentence flesch summary cache txt iajs-2801 Awad, Wisam K.; Kareem, Fatema F. The Homomorphism of Cubic bipolar ideals of a KU-semigroup 2022 11 .pdf application/pdf 6047 293 93 Therefore, eitherπœ‡1 +(0) β‰₯ πœ‡1 +(𝛼),πœ‡1 βˆ’(0) ≀ πœ‡1 βˆ’(𝛼) orπœ‡2 +(0) β‰₯ πœ‡2 +(𝛽),πœ‡2 βˆ’(0) ≀ πœ‡2 βˆ’(𝛽), also, πœ†1 +(0) β‰₯ πœ†1 +(𝛼),πœ†1 βˆ’(0) ≀ πœ†1 βˆ’(𝛼) orπœ†2 +(0) β‰₯ πœ†2 +(𝛽),πœ†2 βˆ’(0) ≀ πœ†2 βˆ’(𝛽) for all 𝛼 , 𝛽 ∈ β„΅. (ii)Suppose that ; πœ‡2 +(0) ≀; πœ‡1 +(𝛼), ; πœ‡2 βˆ’(0) β‰₯; πœ‡1 βˆ’(𝛼) and πœ‡2 +(0) ≀ πœ‡2 +(𝛽),πœ‡2 βˆ’(0) β‰₯ πœ‡2 βˆ’(𝛽) also,; πœ†2 +(0); ≀; ; πœ†1 +(𝛼);,; πœ†2 βˆ’(0); β‰₯; πœ†1 βˆ’(𝛼) and ; πœ†2 +(0) ≀; πœ†2 +(𝛽);,; πœ†2 βˆ’(0); β‰₯; ; πœ†2 βˆ’(𝛽), for all𝛼 , 𝛽 ∈ β„΅. Then (πœ‡1 + Γ— πœ‡2 +)(0,0) = π‘Ÿπ‘šπ‘–π‘›{πœ‡1 +(0), πœ‡2 +(0)} = πœ‡2 +(0) And (πœ‡1 + Γ— πœ‡2 +)(𝛼, 𝛽) = π‘Ÿπ‘šπ‘–π‘›{οΏ½ΜƒοΏ½1 +(𝛼), πœ‡2 +(𝛽)} β‰₯ {πœ‡2 +(0), πœ‡2 +(0)} = πœ‡2 +(0) = (οΏ½ΜƒοΏ½1 + Γ— πœ‡2 +)(0,0) And (πœ‡1 βˆ’ Γ— πœ‡2 βˆ’)(0,0) = π‘Ÿπ‘šπ‘Žπ‘₯{πœ‡1 βˆ’(0), πœ‡2 βˆ’(0)} = πœ‡2 βˆ’(0). (4) Since Ω𝑓1 Γ— Ω𝑓2 is ACB ideal of  , then ; (; οΏ½ΜƒοΏ½1 + Γ—; οΏ½ΜƒοΏ½2 + )(𝛽 1 , 𝛽 2 ); β‰₯ π‘Ÿπ‘šπ‘–π‘›{(; οΏ½ΜƒοΏ½1 + Γ—; οΏ½ΜƒοΏ½2 +)((𝛼1, 𝛼2) βˆ— (𝛽 1 , 𝛽 2 )), (οΏ½ΜƒοΏ½1 + Γ— οΏ½ΜƒοΏ½2 +)(𝛼1, 𝛼2)} = π‘Ÿπ‘šπ‘–π‘›{(οΏ½ΜƒοΏ½1 + Γ— οΏ½ΜƒοΏ½2 + )(𝛼1 βˆ— 𝛽 1 , 𝛼2 βˆ— 𝛽 2 ), (οΏ½ΜƒοΏ½1 + Γ— οΏ½ΜƒοΏ½2 + )(𝛼1, 𝛼2)} Put𝛼1 = 𝛽1 = 0 , then we have (οΏ½ΜƒοΏ½1 + Γ— οΏ½ΜƒοΏ½2 + )(0, 𝛽 2 ) β‰₯ π‘Ÿπ‘šπ‘–π‘›{(; οΏ½ΜƒοΏ½1 + Γ—; οΏ½ΜƒοΏ½2 +)((0, 𝛼2) βˆ— (0, 𝛽 2 )), (; οΏ½ΜƒοΏ½1 + Γ—; οΏ½ΜƒοΏ½2 +)(0, 𝛼2)} = π‘Ÿπ‘šπ‘–π‘›{(; οΏ½ΜƒοΏ½1 + Γ—; οΏ½ΜƒοΏ½2 + )(0, 𝛼2 βˆ— 𝛽 2 ), (; οΏ½ΜƒοΏ½1 + Γ—; οΏ½ΜƒοΏ½2 + )(0, 𝛼2)} and by equation (1), then πœ‡2 +(𝛽2) β‰₯ π‘Ÿπ‘šπ‘–π‘›{οΏ½ΜƒοΏ½2 +(𝛼2 βˆ— 𝛽2), πœ‡2 +(𝛼2)}. cache/iajs-2801.pdf txt/iajs-2801.txt