Residuation in orthomodular lattices
We show that every idempotent weakly divisible residuated lattice satisfying the double negation law can be transformed into an orthomodular lattice. The converse holds if adjointness is replaced by conditional adjointness. Moreover, we show that every positive right residuated lattice satisfying th...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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De Gruyter
2017-04-01
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| Series: | Topological Algebra and its Applications |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/taa-2017-0001 |
| _version_ | 1819291031584112640 |
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| author | Chajda Ivan Länger Helmut |
| author_facet | Chajda Ivan Länger Helmut |
| author_sort | Chajda Ivan |
| collection | DOAJ |
| description | We show that every idempotent weakly divisible residuated lattice satisfying the double negation law can be transformed into an orthomodular lattice. The converse holds if adjointness is replaced by conditional adjointness. Moreover, we show that every positive right residuated lattice satisfying the double negation law and two further simple identities can be converted into an orthomodular lattice. In this case, also the converse statement is true and the corresponence is nearly one-to-one. |
| first_indexed | 2024-12-24T03:32:10Z |
| format | Article |
| id | doaj.art-4b6469b1bde94f968674584209b2805a |
| institution | Directory Open Access Journal |
| issn | 2299-3231 |
| language | English |
| last_indexed | 2024-12-24T03:32:10Z |
| publishDate | 2017-04-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Topological Algebra and its Applications |
| spelling | doaj.art-4b6469b1bde94f968674584209b2805a2022-12-21T17:17:09ZengDe GruyterTopological Algebra and its Applications2299-32312017-04-01511510.1515/taa-2017-0001taa-2017-0001Residuation in orthomodular latticesChajda Ivan0Länger Helmut1Department of Algebra and Geometry, Palacký University Olomouc, 17. listopadu 12, 771 46 Olomouc, Czech RepublicInstitute of Discrete Mathematics and Geometry, TU Wien, Wiedner Hauptstraße 8-10, 1040 Vienna, AustriaWe show that every idempotent weakly divisible residuated lattice satisfying the double negation law can be transformed into an orthomodular lattice. The converse holds if adjointness is replaced by conditional adjointness. Moreover, we show that every positive right residuated lattice satisfying the double negation law and two further simple identities can be converted into an orthomodular lattice. In this case, also the converse statement is true and the corresponence is nearly one-to-one.https://doi.org/10.1515/taa-2017-0001residuated latticeright residuated latticeweak residuated latticeorthomodular latticeweak divisiblepositivedouble negation law |
| spellingShingle | Chajda Ivan Länger Helmut Residuation in orthomodular lattices Topological Algebra and its Applications residuated lattice right residuated lattice weak residuated lattice orthomodular lattice weak divisible positive double negation law |
| title | Residuation in orthomodular lattices |
| title_full | Residuation in orthomodular lattices |
| title_fullStr | Residuation in orthomodular lattices |
| title_full_unstemmed | Residuation in orthomodular lattices |
| title_short | Residuation in orthomodular lattices |
| title_sort | residuation in orthomodular lattices |
| topic | residuated lattice right residuated lattice weak residuated lattice orthomodular lattice weak divisible positive double negation law |
| url | https://doi.org/10.1515/taa-2017-0001 |
| work_keys_str_mv | AT chajdaivan residuationinorthomodularlattices AT langerhelmut residuationinorthomodularlattices |