On cogrowth function of algebras and its logarithmical gap
Let $A \cong k\langle X \rangle / I$ be an associative algebra. A finite word over alphabet $X$ is $I$-reducible if its image in $A$ is a $k$-linear combination of length-lexicographically lesser words. An obstruction is a subword-minimal $I$-reducible word. If the number of obstructions is finite t...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Académie des sciences
2021-04-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.170/ |
| _version_ | 1797651560641069056 |
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| author | Kanel-Belov, Alexei Ya. Melnikov, Igor Mitrofanov, Ivan |
| author_facet | Kanel-Belov, Alexei Ya. Melnikov, Igor Mitrofanov, Ivan |
| author_sort | Kanel-Belov, Alexei Ya. |
| collection | DOAJ |
| description | Let $A \cong k\langle X \rangle / I$ be an associative algebra. A finite word over alphabet $X$ is $I$-reducible if its image in $A$ is a $k$-linear combination of length-lexicographically lesser words. An obstruction is a subword-minimal $I$-reducible word. If the number of obstructions is finite then $I$ has a finite Gröbner basis, and the word problem for the algebra is decidable. A cogrowth function is the number of obstructions of length $\le n$. We show that the cogrowth function of a finitely presented algebra is either bounded or at least logarithmical. We also show that an uniformly recurrent word has at least logarithmical cogrowth. |
| first_indexed | 2024-03-11T16:17:35Z |
| format | Article |
| id | doaj.art-b7b1ccb6addc4fc0bc68288bccabe84a |
| institution | Directory Open Access Journal |
| issn | 1778-3569 |
| language | English |
| last_indexed | 2024-03-11T16:17:35Z |
| publishDate | 2021-04-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj.art-b7b1ccb6addc4fc0bc68288bccabe84a2023-10-24T14:18:42ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692021-04-01359329730310.5802/crmath.17010.5802/crmath.170On cogrowth function of algebras and its logarithmical gapKanel-Belov, Alexei Ya.0Melnikov, Igor1Mitrofanov, Ivan2Bar Ilan University, Ramat-Gan, IsraelMoscow Institute of Physics and Technology, Dolgoprudny, RussiaC.N.R.S., École Normale Superieur, PSL Research University, FranceLet $A \cong k\langle X \rangle / I$ be an associative algebra. A finite word over alphabet $X$ is $I$-reducible if its image in $A$ is a $k$-linear combination of length-lexicographically lesser words. An obstruction is a subword-minimal $I$-reducible word. If the number of obstructions is finite then $I$ has a finite Gröbner basis, and the word problem for the algebra is decidable. A cogrowth function is the number of obstructions of length $\le n$. We show that the cogrowth function of a finitely presented algebra is either bounded or at least logarithmical. We also show that an uniformly recurrent word has at least logarithmical cogrowth.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.170/ |
| spellingShingle | Kanel-Belov, Alexei Ya. Melnikov, Igor Mitrofanov, Ivan On cogrowth function of algebras and its logarithmical gap Comptes Rendus. Mathématique |
| title | On cogrowth function of algebras and its logarithmical gap |
| title_full | On cogrowth function of algebras and its logarithmical gap |
| title_fullStr | On cogrowth function of algebras and its logarithmical gap |
| title_full_unstemmed | On cogrowth function of algebras and its logarithmical gap |
| title_short | On cogrowth function of algebras and its logarithmical gap |
| title_sort | on cogrowth function of algebras and its logarithmical gap |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.170/ |
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