Commutative Rings Behind Divisible Residuated Lattices
Divisible residuated lattices are algebraic structures corresponding to a more comprehensive logic than Hajek’s basic logic with an important significance in the study of fuzzy logic. The purpose of this paper is to investigate commutative rings whose lattice of ideals can be equipped with a structu...
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MDPI AG
2024-12-01
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| author | Cristina Flaut Dana Piciu |
| author_facet | Cristina Flaut Dana Piciu |
| author_sort | Cristina Flaut |
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| description | Divisible residuated lattices are algebraic structures corresponding to a more comprehensive logic than Hajek’s basic logic with an important significance in the study of fuzzy logic. The purpose of this paper is to investigate commutative rings whose lattice of ideals can be equipped with a structure of divisible residuated lattice. We show that these rings are multiplication rings. A characterization, additional examples, and their connections to other classes of rings are established. Furthermore, we analyze the structure of divisible residuated lattices using finite commutative rings. From computational considerations, we present an explicit construction of isomorphism classes of divisible residuated lattices (that are not BL-algebras) of small size <i>n</i> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>6</mn></mrow></semantics></math></inline-formula>), and we give summarizing statistics. |
| format | Article |
| id | doaj-art-13750f9398e945b4adad426de4d73eda |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj-art-13750f9398e945b4adad426de4d73eda2024-12-13T16:28:01ZengMDPI AGMathematics2227-73902024-12-011223386710.3390/math12233867Commutative Rings Behind Divisible Residuated LatticesCristina Flaut0Dana Piciu1Faculty of Mathematics and Computer Science, Ovidius University, Bd. Mamaia 124, 900527 Constanţa, RomaniaFaculty of Science, University of Craiova, A.I. Cuza Street, 13, 200585 Craiova, RomaniaDivisible residuated lattices are algebraic structures corresponding to a more comprehensive logic than Hajek’s basic logic with an important significance in the study of fuzzy logic. The purpose of this paper is to investigate commutative rings whose lattice of ideals can be equipped with a structure of divisible residuated lattice. We show that these rings are multiplication rings. A characterization, additional examples, and their connections to other classes of rings are established. Furthermore, we analyze the structure of divisible residuated lattices using finite commutative rings. From computational considerations, we present an explicit construction of isomorphism classes of divisible residuated lattices (that are not BL-algebras) of small size <i>n</i> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>6</mn></mrow></semantics></math></inline-formula>), and we give summarizing statistics.https://www.mdpi.com/2227-7390/12/23/3867multiplication ringidealdivisible residuated lattice |
| spellingShingle | Cristina Flaut Dana Piciu Commutative Rings Behind Divisible Residuated Lattices Mathematics multiplication ring ideal divisible residuated lattice |
| title | Commutative Rings Behind Divisible Residuated Lattices |
| title_full | Commutative Rings Behind Divisible Residuated Lattices |
| title_fullStr | Commutative Rings Behind Divisible Residuated Lattices |
| title_full_unstemmed | Commutative Rings Behind Divisible Residuated Lattices |
| title_short | Commutative Rings Behind Divisible Residuated Lattices |
| title_sort | commutative rings behind divisible residuated lattices |
| topic | multiplication ring ideal divisible residuated lattice |
| url | https://www.mdpi.com/2227-7390/12/23/3867 |
| work_keys_str_mv | AT cristinaflaut commutativeringsbehinddivisibleresiduatedlattices AT danapiciu commutativeringsbehinddivisibleresiduatedlattices |