On Quantum-MV algebras - Part II: Orthomodular Lattices, Softlattices and Widelattices
Orthomodular lattices generalize the Boolean algebras; they have arisen in the study of quantum logic. Quantum-MV algebras were introduced as non-lattice theoretic generalizations of MV algebras and as non-idempotent generalizations of orthomodular lattices.In this paper, we continue the...
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| Format: | Article |
| Language: | English |
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Islamic Azad University, Bandar Abbas Branch
2022-05-01
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| Series: | Transactions on Fuzzy Sets and Systems |
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| Online Access: | https://tfss.journals.iau.ir/article_690286_013793ba0dbc1fe6d907a3bbd20fb8d0.pdf |
| _version_ | 1797827944787214336 |
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| author | Afrodita Iorgulescu |
| author_facet | Afrodita Iorgulescu |
| author_sort | Afrodita Iorgulescu |
| collection | DOAJ |
| description | Orthomodular lattices generalize the Boolean algebras; they have arisen in the study of quantum logic. Quantum-MV algebras were introduced as non-lattice theoretic generalizations of MV algebras and as non-idempotent generalizations of orthomodular lattices.In this paper, we continue the research in the ``world'' of involutive algebras of the form $(A, \odot, {}^-,1)$, with $1^-=0 $, $1$ being the last element. We clarify now some aspects concerning the quantum-MV (QMV) algebras as non-idempotent generalizations of orthomodular lattices.We study in some detail the orthomodular lattices (OMLs) and we introduce and study two generalizations of them, the orthomodular softlattices (OMSLs) and the orthomodular widelattices (OMWLs). We establish systematically connections between OMLs and OMSLs/OMWLs and QMV, pre-MV, metha-MV, orthomodular algebras and ortholattices, orthosoftlattices/orthowidelattices - connections illustrated in 22 Figures. We prove, among others, that the transitive OMLs coincide with the Boolean algebras, that the OMSLs coincide with the OMLs, that the OMLs are included in OMWLs and that the OMWLs are a proper subclass of QMV algebras. The transitive and/or the antisymmetric case is also studied. |
| first_indexed | 2024-04-09T12:56:32Z |
| format | Article |
| id | doaj.art-e8e3eba2797e4279bfa89577089b3e00 |
| institution | Directory Open Access Journal |
| issn | 2821-0131 |
| language | English |
| last_indexed | 2024-04-09T12:56:32Z |
| publishDate | 2022-05-01 |
| publisher | Islamic Azad University, Bandar Abbas Branch |
| record_format | Article |
| series | Transactions on Fuzzy Sets and Systems |
| spelling | doaj.art-e8e3eba2797e4279bfa89577089b3e002023-05-13T17:23:31ZengIslamic Azad University, Bandar Abbas BranchTransactions on Fuzzy Sets and Systems2821-01312022-05-011114110.30495/tfss.2022.690286690286On Quantum-MV algebras - Part II: Orthomodular Lattices, Softlattices and WidelatticesAfrodita Iorgulescu0Department of Informatics and Economic Cybernetics, Bucharest University of Economic Studies, Bucharest, RomaniaOrthomodular lattices generalize the Boolean algebras; they have arisen in the study of quantum logic. Quantum-MV algebras were introduced as non-lattice theoretic generalizations of MV algebras and as non-idempotent generalizations of orthomodular lattices.In this paper, we continue the research in the ``world'' of involutive algebras of the form $(A, \odot, {}^-,1)$, with $1^-=0 $, $1$ being the last element. We clarify now some aspects concerning the quantum-MV (QMV) algebras as non-idempotent generalizations of orthomodular lattices.We study in some detail the orthomodular lattices (OMLs) and we introduce and study two generalizations of them, the orthomodular softlattices (OMSLs) and the orthomodular widelattices (OMWLs). We establish systematically connections between OMLs and OMSLs/OMWLs and QMV, pre-MV, metha-MV, orthomodular algebras and ortholattices, orthosoftlattices/orthowidelattices - connections illustrated in 22 Figures. We prove, among others, that the transitive OMLs coincide with the Boolean algebras, that the OMSLs coincide with the OMLs, that the OMLs are included in OMWLs and that the OMWLs are a proper subclass of QMV algebras. The transitive and/or the antisymmetric case is also studied.https://tfss.journals.iau.ir/article_690286_013793ba0dbc1fe6d907a3bbd20fb8d0.pdfm-mel algebram-be algebram-pre-bck algebram-bck algebramv algebraquantum-mv algebrapre-mv algebrametha-mv algebraorthomodular algebraortholatticeorthosoftlatticeorthowidelatticeboolean algebra |
| spellingShingle | Afrodita Iorgulescu On Quantum-MV algebras - Part II: Orthomodular Lattices, Softlattices and Widelattices Transactions on Fuzzy Sets and Systems m-mel algebra m-be algebra m-pre-bck algebra m-bck algebra mv algebra quantum-mv algebra pre-mv algebra metha-mv algebra orthomodular algebra ortholattice orthosoftlattice orthowidelattice boolean algebra |
| title | On Quantum-MV algebras - Part II: Orthomodular Lattices, Softlattices and Widelattices |
| title_full | On Quantum-MV algebras - Part II: Orthomodular Lattices, Softlattices and Widelattices |
| title_fullStr | On Quantum-MV algebras - Part II: Orthomodular Lattices, Softlattices and Widelattices |
| title_full_unstemmed | On Quantum-MV algebras - Part II: Orthomodular Lattices, Softlattices and Widelattices |
| title_short | On Quantum-MV algebras - Part II: Orthomodular Lattices, Softlattices and Widelattices |
| title_sort | on quantum mv algebras part ii orthomodular lattices softlattices and widelattices |
| topic | m-mel algebra m-be algebra m-pre-bck algebra m-bck algebra mv algebra quantum-mv algebra pre-mv algebra metha-mv algebra orthomodular algebra ortholattice orthosoftlattice orthowidelattice boolean algebra |
| url | https://tfss.journals.iau.ir/article_690286_013793ba0dbc1fe6d907a3bbd20fb8d0.pdf |
| work_keys_str_mv | AT afroditaiorgulescu onquantummvalgebraspartiiorthomodularlatticessoftlatticesandwidelattices |