On quasi-identities of finite modular lattices. II
The existence of a finite identity basis for any finite lattice was established by R. McKenzie in 1970, but the analogous statement for quasi-identities is incorrect. So, there is a finite lattice that does not have a finite quasi-identity basis and, V.A. Gorbunov and D.M. Smirnov asked which finit...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Academician Ye.A. Buketov Karaganda University
2023-06-01
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| Series: | Қарағанды университетінің хабаршысы. Математика сериясы |
| Subjects: | |
| Online Access: | http://mathematics-vestnik.ksu.kz/index.php/mathematics-vestnik/article/view/565 |
| _version_ | 1797371253742370816 |
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| author | A.O. Basheyeva S.M. Lutsak |
| author_facet | A.O. Basheyeva S.M. Lutsak |
| author_sort | A.O. Basheyeva |
| collection | DOAJ |
| description |
The existence of a finite identity basis for any finite lattice was established by R. McKenzie in 1970, but the analogous statement for quasi-identities is incorrect. So, there is a finite lattice that does not have a finite quasi-identity basis and, V.A. Gorbunov and D.M. Smirnov asked which finite lattices have finite quasiidentity bases. In 1984 V.I. Tumanov conjectured that a proper quasivariety generated by a finite modular lattice is not finitely based. He also found two conditions for quasivarieties which provide this conjecture. We construct a finite modular lattice that does not satisfy Tumanov’s conditions but quasivariety generated by this lattice is not finitely based.
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| first_indexed | 2024-03-08T18:17:15Z |
| format | Article |
| id | doaj.art-04b8f654930a4511aa972b16b0d516ae |
| institution | Directory Open Access Journal |
| issn | 2518-7929 2663-5011 |
| language | English |
| last_indexed | 2024-03-08T18:17:15Z |
| publishDate | 2023-06-01 |
| publisher | Academician Ye.A. Buketov Karaganda University |
| record_format | Article |
| series | Қарағанды университетінің хабаршысы. Математика сериясы |
| spelling | doaj.art-04b8f654930a4511aa972b16b0d516ae2023-12-31T10:28:16ZengAcademician Ye.A. Buketov Karaganda UniversityҚарағанды университетінің хабаршысы. Математика сериясы2518-79292663-50112023-06-01110210.31489/2023m2/45-52On quasi-identities of finite modular lattices. IIA.O. BasheyevaS.M. Lutsak The existence of a finite identity basis for any finite lattice was established by R. McKenzie in 1970, but the analogous statement for quasi-identities is incorrect. So, there is a finite lattice that does not have a finite quasi-identity basis and, V.A. Gorbunov and D.M. Smirnov asked which finite lattices have finite quasiidentity bases. In 1984 V.I. Tumanov conjectured that a proper quasivariety generated by a finite modular lattice is not finitely based. He also found two conditions for quasivarieties which provide this conjecture. We construct a finite modular lattice that does not satisfy Tumanov’s conditions but quasivariety generated by this lattice is not finitely based. http://mathematics-vestnik.ksu.kz/index.php/mathematics-vestnik/article/view/565latticefinite latticemodular latticequasivarietyvarietyquasi-identity |
| spellingShingle | A.O. Basheyeva S.M. Lutsak On quasi-identities of finite modular lattices. II Қарағанды университетінің хабаршысы. Математика сериясы lattice finite lattice modular lattice quasivariety variety quasi-identity |
| title | On quasi-identities of finite modular lattices. II |
| title_full | On quasi-identities of finite modular lattices. II |
| title_fullStr | On quasi-identities of finite modular lattices. II |
| title_full_unstemmed | On quasi-identities of finite modular lattices. II |
| title_short | On quasi-identities of finite modular lattices. II |
| title_sort | on quasi identities of finite modular lattices ii |
| topic | lattice finite lattice modular lattice quasivariety variety quasi-identity |
| url | http://mathematics-vestnik.ksu.kz/index.php/mathematics-vestnik/article/view/565 |
| work_keys_str_mv | AT aobasheyeva onquasiidentitiesoffinitemodularlatticesii AT smlutsak onquasiidentitiesoffinitemodularlatticesii |