A Novel Algebraic Structure of (<i>α</i>,<i>β</i>)-Complex Fuzzy Subgroups
A complex fuzzy set is a vigorous framework to characterize novel machine learning algorithms. This set is more suitable and flexible compared to fuzzy sets, intuitionistic fuzzy sets, and bipolar fuzzy sets. On the aspects of complex fuzzy sets, we initiate the abstraction of <inline-formula>...
| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-07-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/23/8/992 |
| _version_ | 1797523930242613248 |
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| author | Hanan Alolaiyan Halimah A. Alshehri Muhammad Haris Mateen Dragan Pamucar Muhammad Gulzar |
| author_facet | Hanan Alolaiyan Halimah A. Alshehri Muhammad Haris Mateen Dragan Pamucar Muhammad Gulzar |
| author_sort | Hanan Alolaiyan |
| collection | DOAJ |
| description | A complex fuzzy set is a vigorous framework to characterize novel machine learning algorithms. This set is more suitable and flexible compared to fuzzy sets, intuitionistic fuzzy sets, and bipolar fuzzy sets. On the aspects of complex fuzzy sets, we initiate the abstraction of <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>-complex fuzzy sets and then define <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="(" close=")"><mi>α</mi><mo>,</mo><mi>β</mi></mfenced></semantics></math></inline-formula>-complex fuzzy subgroups. Furthermore, we prove that every complex fuzzy subgroup is an <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>-complex fuzzy subgroup and 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>-complex fuzzy normal subgroups of given group. We extend this ideology to 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>-complex fuzzy cosets and analyze some of their algebraic characteristics. Furthermore, we prove that <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>-complex fuzzy normal subgroup is constant in the conjugate classes of group. We present an alternative conceptualization of <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>-complex fuzzy normal subgroup in the sense of the commutator of groups. We establish the <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>-complex fuzzy subgroup of the classical quotient group and show that the set of all <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>-complex fuzzy cosets of this specific complex fuzzy normal subgroup form a group. Additionally, we expound the index of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="(" close=")"><mi>α</mi><mo>,</mo><mi>β</mi></mfenced></semantics></math></inline-formula>-complex fuzzy subgroups and investigate the <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>-complex fuzzification of Lagrange’s theorem analog to Lagrange’ theorem of classical group theory. |
| first_indexed | 2024-03-10T08:50:07Z |
| format | Article |
| id | doaj.art-caed50ade5ae4c1bbd7690737b2ad106 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-10T08:50:07Z |
| publishDate | 2021-07-01 |
| publisher | MDPI AG |
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| series | Entropy |
| spelling | doaj.art-caed50ade5ae4c1bbd7690737b2ad1062023-11-22T07:34:43ZengMDPI AGEntropy1099-43002021-07-0123899210.3390/e23080992A Novel Algebraic Structure of (<i>α</i>,<i>β</i>)-Complex Fuzzy SubgroupsHanan Alolaiyan0Halimah A. Alshehri1Muhammad Haris Mateen2Dragan Pamucar3Muhammad Gulzar4Department of Mathematics, King Saud University, Riyadh 11451, Saudi ArabiaDepartment of Computer Science and Engineering, King Saud University, Riyadh 11451, Saudi ArabiaDepartment of Mathematics, University of the Punjab, Lahore 54590, PakistanDepartment of Logistics, Military Academy, University of Defence in Belgrade, 11000 Belgrade, SerbiaDepartment of Mathematics, Government College University Faisalabad, Faisalabad 38000, PakistanA complex fuzzy set is a vigorous framework to characterize novel machine learning algorithms. This set is more suitable and flexible compared to fuzzy sets, intuitionistic fuzzy sets, and bipolar fuzzy sets. On the aspects of complex fuzzy sets, we initiate the abstraction of <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>-complex fuzzy sets and then define <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="(" close=")"><mi>α</mi><mo>,</mo><mi>β</mi></mfenced></semantics></math></inline-formula>-complex fuzzy subgroups. Furthermore, we prove that every complex fuzzy subgroup is an <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>-complex fuzzy subgroup and 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>-complex fuzzy normal subgroups of given group. We extend this ideology to 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>-complex fuzzy cosets and analyze some of their algebraic characteristics. Furthermore, we prove that <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>-complex fuzzy normal subgroup is constant in the conjugate classes of group. We present an alternative conceptualization of <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>-complex fuzzy normal subgroup in the sense of the commutator of groups. We establish the <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>-complex fuzzy subgroup of the classical quotient group and show that the set of all <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>-complex fuzzy cosets of this specific complex fuzzy normal subgroup form a group. Additionally, we expound the index of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="(" close=")"><mi>α</mi><mo>,</mo><mi>β</mi></mfenced></semantics></math></inline-formula>-complex fuzzy subgroups and investigate the <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>-complex fuzzification of Lagrange’s theorem analog to Lagrange’ theorem of classical group theory.https://www.mdpi.com/1099-4300/23/8/992complex fuzzy set(<i>α</i>,<i>β</i>)-complex fuzzy set(<i>α</i>,<i>β</i>)-complex fuzzy subgroup(<i>α</i>,<i>β</i>)-complex fuzzy normal subgroup |
| spellingShingle | Hanan Alolaiyan Halimah A. Alshehri Muhammad Haris Mateen Dragan Pamucar Muhammad Gulzar A Novel Algebraic Structure of (<i>α</i>,<i>β</i>)-Complex Fuzzy Subgroups Entropy complex fuzzy set (<i>α</i>,<i>β</i>)-complex fuzzy set (<i>α</i>,<i>β</i>)-complex fuzzy subgroup (<i>α</i>,<i>β</i>)-complex fuzzy normal subgroup |
| title | A Novel Algebraic Structure of (<i>α</i>,<i>β</i>)-Complex Fuzzy Subgroups |
| title_full | A Novel Algebraic Structure of (<i>α</i>,<i>β</i>)-Complex Fuzzy Subgroups |
| title_fullStr | A Novel Algebraic Structure of (<i>α</i>,<i>β</i>)-Complex Fuzzy Subgroups |
| title_full_unstemmed | A Novel Algebraic Structure of (<i>α</i>,<i>β</i>)-Complex Fuzzy Subgroups |
| title_short | A Novel Algebraic Structure of (<i>α</i>,<i>β</i>)-Complex Fuzzy Subgroups |
| title_sort | novel algebraic structure of i α i i β i complex fuzzy subgroups |
| topic | complex fuzzy set (<i>α</i>,<i>β</i>)-complex fuzzy set (<i>α</i>,<i>β</i>)-complex fuzzy subgroup (<i>α</i>,<i>β</i>)-complex fuzzy normal subgroup |
| url | https://www.mdpi.com/1099-4300/23/8/992 |
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