A Certain Structure of Bipolar Fuzzy Subrings
The role of symmetry in ring theory is universally recognized. The most directly definable universal relation in a symmetric set theory is isomorphism. This article develops a certain structure of bipolar fuzzy subrings, including bipolar fuzzy quotient ring, bipolar fuzzy ring homomorphism, and bip...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-08-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/13/8/1397 |
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| author | Hanan Alolaiyan Muhammad Haris Mateen Dragan Pamucar Muhammad Khalid Mahmmod Farrukh Arslan |
| author_facet | Hanan Alolaiyan Muhammad Haris Mateen Dragan Pamucar Muhammad Khalid Mahmmod Farrukh Arslan |
| author_sort | Hanan Alolaiyan |
| collection | DOAJ |
| description | The role of symmetry in ring theory is universally recognized. The most directly definable universal relation in a symmetric set theory is isomorphism. This article develops a certain structure of bipolar fuzzy subrings, including bipolar fuzzy quotient ring, bipolar fuzzy ring homomorphism, and bipolar fuzzy ring isomorphism. We define <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo></mrow></semantics></math></inline-formula>-cut of bipolar fuzzy set and investigate the algebraic attributions of this phenomenon. We also define the support set of bipolar fuzzy set and prove various important properties relating to this concept. Additionally, we define bipolar fuzzy homomorphism by using the notion of natural ring homomorphism. We also establish a bipolar fuzzy homomorphism between bipolar fuzzy subring of the quotient ring and bipolar fuzzy subring of this ring. We constituted a significant relationship between two bipolar fuzzy subrings of quotient rings under a given bipolar fuzzy surjective homomorphism. We present the construction of an induced bipolar fuzzy isomorphism between two related bipolar fuzzy subrings. Moreover, to discuss the symmetry between two bipolar fuzzy subrings, we present three fundamental theorems of bipolar fuzzy isomorphism. |
| first_indexed | 2024-03-10T08:20:10Z |
| format | Article |
| id | doaj.art-18f8406f24f24bb684916ef3fb6b62ec |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T08:20:10Z |
| publishDate | 2021-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-18f8406f24f24bb684916ef3fb6b62ec2023-11-22T10:00:47ZengMDPI AGSymmetry2073-89942021-08-01138139710.3390/sym13081397A Certain Structure of Bipolar Fuzzy SubringsHanan Alolaiyan0Muhammad Haris Mateen1Dragan Pamucar2Muhammad Khalid Mahmmod3Farrukh Arslan4Department of Mathematics, King Saud University, Riyadh 11451, Saudi ArabiaDepartment of Mathematics, University of the Punjab, Lahore 54590, PakistanDepartment of Logistics, Military Academy, University of Defence in Belgarde, 11000 Belgarde, SerbiaDepartment of Mathematics, University of the Punjab, Lahore 54590, PakistanDepartment of Electrical Engineering, University of Engineering and Technology, Lahore 54890, PakistanThe role of symmetry in ring theory is universally recognized. The most directly definable universal relation in a symmetric set theory is isomorphism. This article develops a certain structure of bipolar fuzzy subrings, including bipolar fuzzy quotient ring, bipolar fuzzy ring homomorphism, and bipolar fuzzy ring isomorphism. We define <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo></mrow></semantics></math></inline-formula>-cut of bipolar fuzzy set and investigate the algebraic attributions of this phenomenon. We also define the support set of bipolar fuzzy set and prove various important properties relating to this concept. Additionally, we define bipolar fuzzy homomorphism by using the notion of natural ring homomorphism. We also establish a bipolar fuzzy homomorphism between bipolar fuzzy subring of the quotient ring and bipolar fuzzy subring of this ring. We constituted a significant relationship between two bipolar fuzzy subrings of quotient rings under a given bipolar fuzzy surjective homomorphism. We present the construction of an induced bipolar fuzzy isomorphism between two related bipolar fuzzy subrings. Moreover, to discuss the symmetry between two bipolar fuzzy subrings, we present three fundamental theorems of bipolar fuzzy isomorphism.https://www.mdpi.com/2073-8994/13/8/1397bipolar fuzzy setbipolar fuzzy subringbipolar fuzzy idealbipolar fuzzy homomorphismbipolar fuzzy isomorphism |
| spellingShingle | Hanan Alolaiyan Muhammad Haris Mateen Dragan Pamucar Muhammad Khalid Mahmmod Farrukh Arslan A Certain Structure of Bipolar Fuzzy Subrings Symmetry bipolar fuzzy set bipolar fuzzy subring bipolar fuzzy ideal bipolar fuzzy homomorphism bipolar fuzzy isomorphism |
| title | A Certain Structure of Bipolar Fuzzy Subrings |
| title_full | A Certain Structure of Bipolar Fuzzy Subrings |
| title_fullStr | A Certain Structure of Bipolar Fuzzy Subrings |
| title_full_unstemmed | A Certain Structure of Bipolar Fuzzy Subrings |
| title_short | A Certain Structure of Bipolar Fuzzy Subrings |
| title_sort | certain structure of bipolar fuzzy subrings |
| topic | bipolar fuzzy set bipolar fuzzy subring bipolar fuzzy ideal bipolar fuzzy homomorphism bipolar fuzzy isomorphism |
| url | https://www.mdpi.com/2073-8994/13/8/1397 |
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