On fuzzy sub-semi-rings of nexuses

Vajiheh Nazemi Niya; Hojat Babaei; Akbar Rezaei; Vajiheh Nazemi Niya; Hojat Babaei; Akbar Rezaei

doi:10.3934/math.20241715

AIMS Mathematics

2024, Volume 9, Issue 12: 36140-36157. doi: 10.3934/math.20241715

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On fuzzy sub-semi-rings of nexuses

1.
Department of Mathematics, Islamic Azad University of Kerman, Kerman, Iran
2.
Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran

Received: 09 September 2024 Revised: 21 November 2024 Accepted: 06 December 2024 Published: 26 December 2024
MSC : 06F35, 16Y60

In this paper, we first constructed a semi-ring on a nexus and then defined a fuzzy sub-semi-ring associated with a nexus $ N $. We investigated some properties and applications. Fuzzy versions of some well-known crisp concepts are provided over a nexus. We verified some applications of this fuzzy on semi-ring $ N $. We obtained some relationships between sub-semi-ring and fuzzy sub-semi-ring of $ N $. However, these relationships were not true for ideals. We put a condition on fuzzy sub-semi-ring so that these relationships were true for ideals. We defined strong fuzzy sub-semi-ring on $ N $. For strong fuzzy sub-semi-ring on $ N $ and for every $ \alpha\in[0, \mu(0)] $, the level set $ \mu^\alpha $ was an ideal of $ N $. For some strong fuzzy sub-semi-rings $ \mu $, we verified when $ \mu^\alpha $ was a prime ideal of $ N $. In the following, for a semi-ring homomorphism $ f:N\longrightarrow M $, we showed that if $ \mu\in FSUB_S(N) $, then $ f(\mu)\in FSUB_S(M) $ and if $ \mu\in FSUB_S(M) $ then $ f \circ\mu\in FSUB_S(N) $. Finally, we verified some concepts of fuzzy quotient of a nexus semi-ring.
- nexus,
- semi-ring,
- fuzzy sub-semi-ring,
- semi-ring homomorphism,
- fuzzy quotient
Citation: Vajiheh Nazemi Niya, Hojat Babaei, Akbar Rezaei. On fuzzy sub-semi-rings of nexuses[J]. AIMS Mathematics, 2024, 9(12): 36140-36157. doi: 10.3934/math.20241715

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Abstract

In this paper, we first constructed a semi-ring on a nexus and then defined a fuzzy sub-semi-ring associated with a nexus $ N $. We investigated some properties and applications. Fuzzy versions of some well-known crisp concepts are provided over a nexus. We verified some applications of this fuzzy on semi-ring $ N $. We obtained some relationships between sub-semi-ring and fuzzy sub-semi-ring of $ N $. However, these relationships were not true for ideals. We put a condition on fuzzy sub-semi-ring so that these relationships were true for ideals. We defined strong fuzzy sub-semi-ring on $ N $. For strong fuzzy sub-semi-ring on $ N $ and for every $ \alpha\in[0, \mu(0)] $, the level set $ \mu^\alpha $ was an ideal of $ N $. For some strong fuzzy sub-semi-rings $ \mu $, we verified when $ \mu^\alpha $ was a prime ideal of $ N $. In the following, for a semi-ring homomorphism $ f:N\longrightarrow M $, we showed that if $ \mu\in FSUB_S(N) $, then $ f(\mu)\in FSUB_S(M) $ and if $ \mu\in FSUB_S(M) $ then $ f \circ\mu\in FSUB_S(N) $. Finally, we verified some concepts of fuzzy quotient of a nexus semi-ring.

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