| Тождества пространств линейных преобразований и неассоциативных линейных алгебр | |
| Кислицын А. В.1 | |
| 1.1Омский государственный университет им. Ф. М. Достоевского, | |
| Дата поступления 2024.10.14 | Аннотация. В работе представлены основные результаты о мультипликативных векторных пространствах, тождествах мультипликативных векторных пространств и $L$-многообразиях. Также приведены следствия полученных результатов для неассоциативных линейных алгебр |
| Ключевые слова мультипликативное векторное пространство, тождество мультипликативного векторного пространства, базис тождеств, конечно базируемое пространство, не конечно базируемое пространство, $L$-многообразие, существенно бесконечно базируемое многообразие, сильно б | |
Библиография \bibitem{kislitsin-10} Kislitsin~A.V. On identities of spaces of linear transformations over infinite field~// The News of Altai State University. 2010. No.\,1/2\,(65). Pp.~37--41. (In Russian). \bibitem{lvov-78} L'vov~I.V. Finite-dimensional algebras with infinite bases of identities~// Siberian Mathematical Journal. 1978. Vol.\,19, no.\,1. Pp.~63--69. \bibitem{razmyslov-73} Razmyslov~Yu.P. The existence of a finite basis for the identities of the matrix algebra of order two over a field of characteristic zero~// Algebra Logika. Vol.\,12, no.\,1. Pp.~83--113. \bibitem{specht-50} Specht~W. Gesetze in Ringen, I~// Mathematische Zeitschrift. 1950. Vol.~52, no.\,5. Pp.~557--589. \bibitem{kemer-87} Kemer~A.R. Finite basis property of identities of associative algebras~// Algebra i Logika. 1987. Vol.\,26, no.\,5. Pp.~362--397. \bibitem{belov-10} Belov~A.Ya. Local finite basis property and local representability of varieties of associative rings~// Izvestiya: Mathematics. 2010. Vol.\,74, no.\,1. Pp.~1--126. \bibitem{lyndon-54} Lyndon~R.C. Identities in finite algebras~// Proceedings of the American Mathematical Society. 1954. Vol.\,5, no.\,1. Pp.~8--9. \bibitem{perkins-69} Perkins~P. Bases of equational theories of semigroups~// Journal~of Algebra. 1969. Vol.\,11, no.\,2. Pp.~293--314. \bibitem{polin-76} Polin~S.V. Identities of finite algebras~// Siberian Mathematical Journal. 1976. Vol.\,17, no.\,6. Pp.~992--999. \bibitem{lee-79} Vaughan-Lee~M.R. Laws in finite loops~// Algebra Universalis. 1979. Vol.\,9, no.\,3. Pp.~269--280. \bibitem{iskis-13} Isaev~I.M., Kislitsin~A.V. Identities in vector spaces and examples of finite-dimensional linear algebras having no finite basis of identities~// Algebra and Logic. 2013. Vol.\,52, no.\,4. Pp.~290--307. \bibitem{isaev-89} Isaev~I.M. Essentially non finite based varieties of algebras~// Siberian Mathematical Journal. 1989. Vol.\,30, no.\,6. Pp.~892--894. \bibitem{iskis-15} Isaev~I.M., Kislitsin~A.V. Identities of vector spaces embedded in linear algebras~// Siberian Electronic Mathematical Reports. 2015. Vol.\,12. Pp.~328--342. (In Russian). \bibitem{iskis-17} Isaev~I.M., Kislitsin~A.V. Identities in vector spaces embedded in finite associative algebras~// Journal of Mathematical Sciences. 2017. Vol.\,221, no.\,6. Pp.~849--856. \bibitem{sapir-88} Sapir~M.V. Inherently nonfinitely based finite semigroups~// Mathematics of the USSR-Sbornik. 1988. Vol.\,61, no.\,1. Pp.~155--166. \bibitem{kislitsin-18} Kislitsin~A.V. The Specht property of $L$-varieties of vector spaces over an arbitrary field~// Algebra and Logic. 2018. Vol.\,57, no.\,5. Pp.~556--566. \bibitem{drensky-21} Drensky~V.S. Weak polynomial identities and their applications~// Communications in Mathematics. 2021. Vol.\,29, no.\,2. Pp.~291--324. \bibitem{kislitsin-22-1} Kislitsin~A.V. The Specht property of $L$-varieties of vector spaces over an arbitrary field~// Second Conference of Russian Mathematical Centers (November 7--11, 2022): collection of abstracts. Moscow: Moscow University Publishing House, 2022. Pp.~112--114. \bibitem{maltsev-76} Maltsev~Yu.N. Almost commutative varieties of associative rings~// Siberian Mathematical Journal. 1976. Vol.\,17, no.\,5. Pp.~803--811. \bibitem{maltsev-96} Maltsev~Yu.N. Just non commutative varieties of operator algebras and rings with some conditions on nilpotent elements~// Tamkang Journal Mathematics. 1996. Vol.\,17, no.\,1. Pp.~473--496. \bibitem{finogenova-04} Finogenova~O.B. Varieties of associative algebras satisfying Engel identities~// Algebra and Logic. 2004. Vol.\,43, no.\,4. Pp.~271--284. \bibitem{finogenova-13} Finogenova~O.B. Almost commutative varieties of associative rings and algebras over a finite field~// Algebra and Logic. 2013. Vol.\,52, no.\,6. Pp.~484--510. \bibitem{kislitsin-18-1} Kislitsin~A.V. On nonnilpotent almost commutative $L$-varieties of vector spaces~// Siberian Mathematical Journal. 2018. Vol.\,59, no.\,3. Pp.~580--586. \bibitem{kislitsin-21} Kislitsin~A.\,V. On almost Engel $L$-varieties of vector spaces~// Siberian Electronic Mathematical Reports. 2021. Vol.\,12, no.\,2. Pp.~1705--1713. (In Russian). \bibitem{tarski-56} Tarski~A. Equationally complete rings and relation algebras~// Proceedings Koninklijke Nederlandse Akademie van Wetenschappen. 1956. Vol.\,59, no.\,1. Pp.~39--46. \bibitem{kislitsin-22} Kislitsin~A.V. Minimal nonzero $L$-varieties of vector spaces over the field~$\mathbb Z_2$~// Algebra Logika. 2022. Vol.\,61, no.\,4. Pp.~461--468. (In Russian). \bibitem{malpar-77} Maltsev~Yu.N., Parfenov~V.A. An example of a nonassociative algebra not admitting a finite basis of identities~// Siberian Mathematical Journal. 1977. Vol.\,18, no.\,6. Pp.~1420--1421. \bibitem{isaev-18} Isaev~I.M. On the join of Spechtian varieties of algebras~// Siberian Electronic Mathematical Reports. 2021. Vol.\,15. Pp.~1498--1505. (In Russian). \bibitem{dnister-93} Filippov~V.I., Kharchenko~V.K., Shestakov~I.P. Dniester notebook. 4 ed. Novosibirsk: Mathematics Institute, Russian Academy of Sciences, Siberian Branch, 1993. 73\,p. (In Russian). \bibitem{shestakov-11} Shestakov~I., Zaycev~M. Polynomial identities of finite dimensional simple algebras~// Communications in algebra. 2011. Vol.\,39, no.\,3. Pp.~929--932. \bibitem{iskis-12} Isaev~I.M., Kislitsin~A.V. An example of a simple finite-dimensional algebra with no finite basis of identities~// Doklady Mathematics. 2012. Vol.\,86, no.\,3. Pp.~774--775. \bibitem{kislitsin-15} Kislitsin~A.V. An example of a central simple commutative finite-dimensional algebra with an infinite basis of identities~// Algebra and Logic. 2015. Vol.\,54, no.\,5. Pp.~204--210. \bibitem{kislitsin-17} Kislitsin~A.V. Simple finite-dimensional algebras without finite basis of identities~// Siberian Mathematical Journal. 2017. Vol.\,58, no.\,3. Pp.~461--466. \bibitem{kislitsin-23} Kislitsin~A.V. On simple finite-dimensional algebras with infinite basis of its identities~// Mathematical Notes. 2024. Vol.~114, no.~5--6. Pp.~845--849. | |
Сведения о финансировании и благодарности The author expresses deep appreciation to Prof. V.\,A.~Romankov for his attention to the paper. Supported by Russian Science Foundation, project No. 22-21-00745. | |
| Identities of Spaces of Linear Transformations and Nonassociative Linear~Algebras | |
| Kislitsin A. V.1 | |
| 1.1Dostoevsky Omsk State University | |
| Received 2024.10.14 | Abstract. The paper presents the main results on multiplicative vector spaces, identities of multiplicative vector spaces and $L$-manifolds. The consequences of the obtained results for nonassociative linear algebras are also given |
| Keywords multiplicative vector space, identity of multiplicative vector space, basis of identities, finitely based space, not finitely based space, $L$-manifold, essentially infinitely based manifold, strongly infinitely based manifold | |
References \bibitem{kislitsin-10} Kislitsin~A.V. On identities of spaces of linear transformations over infinite field~// The News of Altai State University. 2010. No.\,1/2\,(65). Pp.~37--41. (In Russian). \bibitem{lvov-78} L'vov~I.V. Finite-dimensional algebras with infinite bases of identities~// Siberian Mathematical Journal. 1978. Vol.\,19, no.\,1. Pp.~63--69. \bibitem{razmyslov-73} Razmyslov~Yu.P. The existence of a finite basis for the identities of the matrix algebra of order two over a field of characteristic zero~// Algebra Logika. Vol.\,12, no.\,1. Pp.~83--113. \bibitem{specht-50} Specht~W. Gesetze in Ringen, I~// Mathematische Zeitschrift. 1950. Vol.~52, no.\,5. Pp.~557--589. \bibitem{kemer-87} Kemer~A.R. Finite basis property of identities of associative algebras~// Algebra i Logika. 1987. Vol.\,26, no.\,5. Pp.~362--397. \bibitem{belov-10} Belov~A.Ya. Local finite basis property and local representability of varieties of associative rings~// Izvestiya: Mathematics. 2010. Vol.\,74, no.\,1. Pp.~1--126. \bibitem{lyndon-54} Lyndon~R.C. Identities in finite algebras~// Proceedings of the American Mathematical Society. 1954. Vol.\,5, no.\,1. Pp.~8--9. \bibitem{perkins-69} Perkins~P. Bases of equational theories of semigroups~// Journal~of Algebra. 1969. Vol.\,11, no.\,2. Pp.~293--314. \bibitem{polin-76} Polin~S.V. Identities of finite algebras~// Siberian Mathematical Journal. 1976. Vol.\,17, no.\,6. Pp.~992--999. \bibitem{lee-79} Vaughan-Lee~M.R. Laws in finite loops~// Algebra Universalis. 1979. Vol.\,9, no.\,3. Pp.~269--280. \bibitem{iskis-13} Isaev~I.M., Kislitsin~A.V. Identities in vector spaces and examples of finite-dimensional linear algebras having no finite basis of identities~// Algebra and Logic. 2013. Vol.\,52, no.\,4. Pp.~290--307. \bibitem{isaev-89} Isaev~I.M. Essentially non finite based varieties of algebras~// Siberian Mathematical Journal. 1989. Vol.\,30, no.\,6. Pp.~892--894. \bibitem{iskis-15} Isaev~I.M., Kislitsin~A.V. Identities of vector spaces embedded in linear algebras~// Siberian Electronic Mathematical Reports. 2015. Vol.\,12. Pp.~328--342. (In Russian). \bibitem{iskis-17} Isaev~I.M., Kislitsin~A.V. Identities in vector spaces embedded in finite associative algebras~// Journal of Mathematical Sciences. 2017. Vol.\,221, no.\,6. Pp.~849--856. \bibitem{sapir-88} Sapir~M.V. Inherently nonfinitely based finite semigroups~// Mathematics of the USSR-Sbornik. 1988. Vol.\,61, no.\,1. Pp.~155--166. \bibitem{kislitsin-18} Kislitsin~A.V. The Specht property of $L$-varieties of vector spaces over an arbitrary field~// Algebra and Logic. 2018. Vol.\,57, no.\,5. Pp.~556--566. \bibitem{drensky-21} Drensky~V.S. Weak polynomial identities and their applications~// Communications in Mathematics. 2021. Vol.\,29, no.\,2. Pp.~291--324. \bibitem{kislitsin-22-1} Kislitsin~A.V. The Specht property of $L$-varieties of vector spaces over an arbitrary field~// Second Conference of Russian Mathematical Centers (November 7--11, 2022): collection of abstracts. Moscow: Moscow University Publishing House, 2022. Pp.~112--114. \bibitem{maltsev-76} Maltsev~Yu.N. Almost commutative varieties of associative rings~// Siberian Mathematical Journal. 1976. Vol.\,17, no.\,5. Pp.~803--811. \bibitem{maltsev-96} Maltsev~Yu.N. Just non commutative varieties of operator algebras and rings with some conditions on nilpotent elements~// Tamkang Journal Mathematics. 1996. Vol.\,17, no.\,1. Pp.~473--496. \bibitem{finogenova-04} Finogenova~O.B. Varieties of associative algebras satisfying Engel identities~// Algebra and Logic. 2004. Vol.\,43, no.\,4. Pp.~271--284. \bibitem{finogenova-13} Finogenova~O.B. Almost commutative varieties of associative rings and algebras over a finite field~// Algebra and Logic. 2013. Vol.\,52, no.\,6. Pp.~484--510. \bibitem{kislitsin-18-1} Kislitsin~A.V. On nonnilpotent almost commutative $L$-varieties of vector spaces~// Siberian Mathematical Journal. 2018. Vol.\,59, no.\,3. Pp.~580--586. \bibitem{kislitsin-21} Kislitsin~A.\,V. On almost Engel $L$-varieties of vector spaces~// Siberian Electronic Mathematical Reports. 2021. Vol.\,12, no.\,2. Pp.~1705--1713. (In Russian). \bibitem{tarski-56} Tarski~A. Equationally complete rings and relation algebras~// Proceedings Koninklijke Nederlandse Akademie van Wetenschappen. 1956. Vol.\,59, no.\,1. Pp.~39--46. \bibitem{kislitsin-22} Kislitsin~A.V. Minimal nonzero $L$-varieties of vector spaces over the field~$\mathbb Z_2$~// Algebra Logika. 2022. Vol.\,61, no.\,4. Pp.~461--468. (In Russian). \bibitem{malpar-77} Maltsev~Yu.N., Parfenov~V.A. An example of a nonassociative algebra not admitting a finite basis of identities~// Siberian Mathematical Journal. 1977. Vol.\,18, no.\,6. Pp.~1420--1421. \bibitem{isaev-18} Isaev~I.M. On the join of Spechtian varieties of algebras~// Siberian Electronic Mathematical Reports. 2021. Vol.\,15. Pp.~1498--1505. (In Russian). \bibitem{dnister-93} Filippov~V.I., Kharchenko~V.K., Shestakov~I.P. Dniester notebook. 4 ed. Novosibirsk: Mathematics Institute, Russian Academy of Sciences, Siberian Branch, 1993. 73\,p. (In Russian). \bibitem{shestakov-11} Shestakov~I., Zaycev~M. Polynomial identities of finite dimensional simple algebras~// Communications in algebra. 2011. Vol.\,39, no.\,3. Pp.~929--932. \bibitem{iskis-12} Isaev~I.M., Kislitsin~A.V. An example of a simple finite-dimensional algebra with no finite basis of identities~// Doklady Mathematics. 2012. Vol.\,86, no.\,3. Pp.~774--775. \bibitem{kislitsin-15} Kislitsin~A.V. An example of a central simple commutative finite-dimensional algebra with an infinite basis of identities~// Algebra and Logic. 2015. Vol.\,54, no.\,5. Pp.~204--210. \bibitem{kislitsin-17} Kislitsin~A.V. Simple finite-dimensional algebras without finite basis of identities~// Siberian Mathematical Journal. 2017. Vol.\,58, no.\,3. Pp.~461--466. \bibitem{kislitsin-23} Kislitsin~A.V. On simple finite-dimensional algebras with infinite basis of its identities~// Mathematical Notes. 2024. Vol.~114, no.~5--6. Pp.~845--849. | |
Acknowledgements The author expresses deep appreciation to Prof. V.\,A.~Romankov for his attention to the paper. Supported by Russian Science Foundation, project No. 22-21-00745. | |
| Сведения об авторах Кислицын А. В. 1.1 |
| About the authors Kislitsin A. V. 1.1 |
