On Γ-Interval Valued Fuzzification of Lagrange’s Theorem of Γ-Interval Valued Fuzzy Subgroups
In this paper, we present the idea of interval valued fuzzy subgroup defined over a certain t-conorm (<inline-formula> <tex-math notation="LaTeX">$\mathrm {\Gamma }$ </tex-math></inline-formula>-IVFSG) and prove that every IVFSG is <inline-formula> <tex-mat...
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IEEE
2020-01-01
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| Series: | IEEE Access |
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| Online Access: | https://ieeexplore.ieee.org/document/9095282/ |
| _version_ | 1811212392775811072 |
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| author | Umer Shuaib Abdul Razaq Hanan Alolaiyan Muhammad Shahram Saif Ayesha Rafiq |
| author_facet | Umer Shuaib Abdul Razaq Hanan Alolaiyan Muhammad Shahram Saif Ayesha Rafiq |
| author_sort | Umer Shuaib |
| collection | DOAJ |
| description | In this paper, we present the idea of interval valued fuzzy subgroup defined over a certain t-conorm (<inline-formula> <tex-math notation="LaTeX">$\mathrm {\Gamma }$ </tex-math></inline-formula>-IVFSG) and prove that every IVFSG is <inline-formula> <tex-math notation="LaTeX">$\mathrm {\Gamma }$ </tex-math></inline-formula>-IVFSG. We use this ideology to define the concepts of <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-IVF cosets, <inline-formula> <tex-math notation="LaTeX">$\mathrm {\Gamma }$ </tex-math></inline-formula>-IVFNSG and formulate their various important algebraic characteristics. We also propose the study of the notion of level subgroups of <inline-formula> <tex-math notation="LaTeX">$\mathrm {\Gamma }$ </tex-math></inline-formula>-IVFSG and investigate the condition under which a <inline-formula> <tex-math notation="LaTeX">$\mathrm {\Gamma }$ </tex-math></inline-formula>-IVFS is <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-IVFSG. Moreover, we extend the study of this phenomenon to introduce the concept of quotient group of a group <inline-formula> <tex-math notation="LaTeX">$Z$ </tex-math></inline-formula> relative to the <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-IVFNSG and acquire a correspondence between each <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-IVF(N)SG of a group <inline-formula> <tex-math notation="LaTeX">$Z$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-IVF(N)SG of its quotient group. Furthermore, we define the index of <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-IVFSG and establish the <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-interval valued fuzzification of Lagrange’s theorem of any <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-IVFSG of a finite group <inline-formula> <tex-math notation="LaTeX">$Z$ </tex-math></inline-formula>. |
| first_indexed | 2024-04-12T05:28:40Z |
| format | Article |
| id | doaj.art-0fc24388153a44daa1806310b73e17b8 |
| institution | Directory Open Access Journal |
| issn | 2169-3536 |
| language | English |
| last_indexed | 2024-04-12T05:28:40Z |
| publishDate | 2020-01-01 |
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| series | IEEE Access |
| spelling | doaj.art-0fc24388153a44daa1806310b73e17b82022-12-22T03:46:11ZengIEEEIEEE Access2169-35362020-01-018962619627010.1109/ACCESS.2020.29955459095282On Γ-Interval Valued Fuzzification of Lagrange’s Theorem of Γ-Interval Valued Fuzzy SubgroupsUmer Shuaib0https://orcid.org/0000-0002-8979-0816Abdul Razaq1https://orcid.org/0000-0002-1898-4082Hanan Alolaiyan2https://orcid.org/0000-0002-7845-7751Muhammad Shahram Saif3Ayesha Rafiq4https://orcid.org/0000-0001-9452-1859Department of Mathematics, Government College University, Faisalabad, PakistanDepartment of Mathematics, Division of Science and Technology, University of Education, Lahore, PakistanDepartment of Mathematics, King Saud University, Riyadh, Saudi ArabiaDepartment of Mathematics, Government College University, Faisalabad, PakistanDepartment of Applied Mathematics and Statistics, Institute of Space Technology, Islamabad, PakistanIn this paper, we present the idea of interval valued fuzzy subgroup defined over a certain t-conorm (<inline-formula> <tex-math notation="LaTeX">$\mathrm {\Gamma }$ </tex-math></inline-formula>-IVFSG) and prove that every IVFSG is <inline-formula> <tex-math notation="LaTeX">$\mathrm {\Gamma }$ </tex-math></inline-formula>-IVFSG. We use this ideology to define the concepts of <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-IVF cosets, <inline-formula> <tex-math notation="LaTeX">$\mathrm {\Gamma }$ </tex-math></inline-formula>-IVFNSG and formulate their various important algebraic characteristics. We also propose the study of the notion of level subgroups of <inline-formula> <tex-math notation="LaTeX">$\mathrm {\Gamma }$ </tex-math></inline-formula>-IVFSG and investigate the condition under which a <inline-formula> <tex-math notation="LaTeX">$\mathrm {\Gamma }$ </tex-math></inline-formula>-IVFS is <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-IVFSG. Moreover, we extend the study of this phenomenon to introduce the concept of quotient group of a group <inline-formula> <tex-math notation="LaTeX">$Z$ </tex-math></inline-formula> relative to the <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-IVFNSG and acquire a correspondence between each <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-IVF(N)SG of a group <inline-formula> <tex-math notation="LaTeX">$Z$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-IVF(N)SG of its quotient group. Furthermore, we define the index of <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-IVFSG and establish the <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-interval valued fuzzification of Lagrange’s theorem of any <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-IVFSG of a finite group <inline-formula> <tex-math notation="LaTeX">$Z$ </tex-math></inline-formula>.https://ieeexplore.ieee.org/document/9095282/Γ-interval valued fuzzy set (Γ-IVFS)Γ-interval valued fuzzy subgroup (Γ-IVFSG)Γ-interval valued fuzzy cosetΓ-interval valued fuzzy normal subgroup (Γ-IVFNSG)Γ-interval valued fuzzy quotient group |
| spellingShingle | Umer Shuaib Abdul Razaq Hanan Alolaiyan Muhammad Shahram Saif Ayesha Rafiq On Γ-Interval Valued Fuzzification of Lagrange’s Theorem of Γ-Interval Valued Fuzzy Subgroups IEEE Access Γ-interval valued fuzzy set (Γ-IVFS) Γ-interval valued fuzzy subgroup (Γ-IVFSG) Γ-interval valued fuzzy coset Γ-interval valued fuzzy normal subgroup (Γ-IVFNSG) Γ-interval valued fuzzy quotient group |
| title | On Γ-Interval Valued Fuzzification of Lagrange’s Theorem of Γ-Interval Valued Fuzzy Subgroups |
| title_full | On Γ-Interval Valued Fuzzification of Lagrange’s Theorem of Γ-Interval Valued Fuzzy Subgroups |
| title_fullStr | On Γ-Interval Valued Fuzzification of Lagrange’s Theorem of Γ-Interval Valued Fuzzy Subgroups |
| title_full_unstemmed | On Γ-Interval Valued Fuzzification of Lagrange’s Theorem of Γ-Interval Valued Fuzzy Subgroups |
| title_short | On Γ-Interval Valued Fuzzification of Lagrange’s Theorem of Γ-Interval Valued Fuzzy Subgroups |
| title_sort | on x0393 interval valued fuzzification of lagrange x2019 s theorem of x0393 interval valued fuzzy subgroups |
| topic | Γ-interval valued fuzzy set (Γ-IVFS) Γ-interval valued fuzzy subgroup (Γ-IVFSG) Γ-interval valued fuzzy coset Γ-interval valued fuzzy normal subgroup (Γ-IVFNSG) Γ-interval valued fuzzy quotient group |
| url | https://ieeexplore.ieee.org/document/9095282/ |
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