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Rough prime bi-\(\Gamma\)-hyperideals and fuzzy prime bi-\(\Gamma\)-hyperideals of \(\Gamma\)-semihypergroups. (English) Zbl 1499.20201

Summary: In this paper, we introduce the concept of prime bi-\(\Gamma\)-hyperideals, rough prime bi-\(\Gamma\)-hyperideals and fuzzy prime bi-\(\Gamma\)-hyperideals of \(\Gamma\)-semihypergroups. We prove that the lower approximation of a prime bi-\(\Gamma\)-hyperideal is a prime bi-\(\Gamma\)-hyperideal and the upper approximation of a prime bi-\(\Gamma\)-hyperideal is a prime bi-\(\Gamma\)-hyperideal. Also the rough set theory is applied to prime bi-\(\Gamma\)-hyperideals in the quotient \(\Gamma\)-semihypergroups. In the end, the notion of fuzzy prime bi-\(\Gamma\)-hyperideals of \(\Gamma\)-semihypergroups has been introduced, and we proved that a bi-\(\Gamma\)-hyperideal \(B\) of a \(\Gamma\)-semihypergroup \(H\) is prime (resp., strongly prime) if and only if the characteristic function \(\chi_B\) of \(B\) is a fuzzy prime (resp., fuzzy strongly prime) bi-\(\Gamma\)-hyperideal of \(H\).

MSC:

20N25 Fuzzy groups
20N20 Hypergroups
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