id author title date pages extension mime words sentence flesch summary cache txt cana-2746 T. Anitha Characterization of Pythagorean Fuzzy Bi-Interior Ideal and Bi-Quasi-Ideal in Ξ“-Semirings 2024 18 .pdf application/pdf 6060 266 82 Hence πœ’π΅ is a 𝑃𝐹𝐡𝐼𝐼 of 𝑆. Conversely, let us assume that πœ’π΅ is a 𝑃𝐹𝐡𝐼𝐼 of 𝑆. Then 𝐡 is a sub-Ξ“ semiring of 𝑆. We have πœ’π‘† ∘ πœ’π΅ ∘ πœ’π‘† ∩ πœ’π΅ ∘ πœ’π‘† ∘ πœ’π΅ βŠ† πœ’π΅ πœ’π‘†Ξ“π΅Ξ“π‘† ∩ πœ’π΅Ξ“π‘†Ξ“π΅ βŠ† πœ’π΅ πœ’π‘†π΅π‘†βˆ©π΅π‘†π΅ βŠ† πœ’π΅ 𝑆𝐡𝑆 ∩ 𝐡𝑆𝐡 βŠ† 𝐡 Hence πœ’π‘† ∘ π΄πœ‡ ∘ πœ’π‘† ∩ π΄πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡ βŠ‡ π΄πœ‡. cache/cana-2746.pdf txt/cana-2746.txt