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Journals / Black Sea Journal of Engineering and Science / 2026 / Cilt: 9 Sayı: 4

Picture Fuzzy Ideals in Semigroups under the Type-3 Order: Bi-Ideals, Quasi-Ideals, and Interior Ideals

Emine Funda Okumuş(KARADENİZ TEKNİK ÜNİVERSİTESİ, FEN FAKÜLTESİ)0000-0002-2878-8448
Sultan Yamak(KARADENIZ TECHNICAL UNIVERSITY)0000-0001-6499-4436
Pages
1607–1612
Publication Date
2026-07-15
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Finally, we show that the corresponding ideal families form complete lattices and that these structures are preserved under standard homomorphic constructions.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:This paper studies picture fuzzy subsets of semigroups under the order (D^*,≤_3 ).
📊
Key Findings & Data:We first clarify the admissible domain D^*={(φ,ψ,ω)∈[0,1]^3:0≤φ+ψ+ω≤1} and then redefine basic operations (intersection, product, Cartesian product, image, inverse image, and characteristic function) so that all results remain inside the picture fuzzy framework.
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Conclusion & Impact:Finally, we show that the corresponding ideal families form complete lattices and that these structures are preserved under standard homomorphic constructions.
🔬Design:Araştırma Makalesi
📍Scope:Comprehensive Sample
⚙️Variables:Picture fuzzy ideals · Semigroups · Picture fuzzy sets

Abstract

This paper studies picture fuzzy subsets of semigroups under the order (D^*,≤_3 ). We first clarify the admissible domain D^*={(φ,ψ,ω)∈[0,1]^3:0≤φ+ψ+ω≤1} and then redefine basic operations (intersection, product, Cartesian product, image, inverse image, and characteristic function) so that all results remain inside the picture fuzzy framework. Based on these operations, we define picture fuzzy subsemigroups, bi-ideals, quasi-ideals, and interior ideals, and we prove their fundamental closure and characterization properties. We also provide counterexamples showing why some earlier formulations can fail to produce valid picture fuzzy sets. Finally, we show that the corresponding ideal families form complete lattices and that these structures are preserved under standard homomorphic constructions.

Özet

This paper studies picture fuzzy subsets of semigroups under the order (D^*,≤_3 ). We first clarify the admissible domain D^*={(φ,ψ,ω)∈[0,1]^3:0≤φ+ψ+ω≤1} and then redefine basic operations (intersection, product, Cartesian product, image, inverse image, and characteristic function) so that all results remain inside the picture fuzzy framework. Based on these operations, we define picture fuzzy subsemigroups, bi-ideals, quasi-ideals, and interior ideals, and we prove their fundamental closure and characterization properties. We also provide counterexamples showing why some earlier formulations can fail to produce valid picture fuzzy sets. Finally, we show that the corresponding ideal families form complete lattices and that these structures are preserved under standard homomorphic constructions.

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Cite This Article

Okumuş, E. F., Yamak, S. (2026). Picture Fuzzy Ideals in Semigroups under the Type-3 Order: Bi-Ideals, Quasi-Ideals, and Interior Ideals. Black Sea Journal of Engineering and Science, Cilt: 9 Sayı: 4, 1607–1612. https://doi.org/10.34248/bsengineering.1925455

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