Gröbner-Shirshov bases and embeddings of algebras. (English) Zbl 1236.16022
Summary: By using Gröbner-Shirshov bases, we show that in the following classes, each (respectively, countably generated) algebra can be embedded into a simple (respectively, two-generated) algebra: associative differential algebras, associative \(\Omega\)-algebras, associative \(\lambda\)-differential algebras. We show that in the following classes, each countably generated algebra over a countable field \(k\) can be embedded into a simple two-generated algebra: associative algebras, semigroups, Lie algebras, associative differential algebras, associative \(\Omega\)-algebras, associative \(\lambda\)-differential algebras. We give other proofs of the well known theorems: each countably generated group (respectively, associative algebra, semigroup, Lie algebra) can be embedded into a two-generated group (respectively, associative algebra, semigroup, Lie algebra).
MSC:
| 16S15 | Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting) |
| 16S32 | Rings of differential operators (associative algebraic aspects) |
| 17B05 | Structure theory for Lie algebras and superalgebras |
| 20F05 | Generators, relations, and presentations of groups |
| 20M05 | Free semigroups, generators and relations, word problems |
| 08A30 | Subalgebras, congruence relations |
| 13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |
Keywords:
Gröbner-Shirshov bases; associative algebras; Lie algebras; associative differential algebras; associative \(\Omega\)-algebras; countably generated groups; countably generated semigroups; two-generator groupsReferences:
| [1] | DOI: 10.1073/pnas.36.7.372 · Zbl 0037.15904 · doi:10.1073/pnas.36.7.372 |
| [2] | DOI: 10.2140/pjm.1960.10.731 · Zbl 0095.12705 · doi:10.2140/pjm.1960.10.731 |
| [3] | DOI: 10.1016/0001-8708(78)90010-5 · Zbl 0326.16019 · doi:10.1016/0001-8708(78)90010-5 |
| [4] | Bokut L. A., Dokl. Akad. Nauk SSSR 14 pp 963– |
| [5] | Bokut L. A., Algebra i Logika 1 pp 47– |
| [6] | Bokut L. A., Sibirsk. Mat. Zh. 4 pp 729– |
| [7] | Bokut L. A., Izv. Akad. Nauk. SSSR Ser. Mat. 36 pp 1173– |
| [8] | Bokut L. A., Algebra i Logika 15 pp 117– |
| [9] | Bokut L. A., Trudy Mat. Inst. Steklov. 148 pp 30– |
| [10] | Bokut L. A., Southeast Asian Bull. Math. 31 pp 1057– |
| [11] | DOI: 10.1016/j.jpaa.2009.05.005 · Zbl 1213.16014 · doi:10.1016/j.jpaa.2009.05.005 |
| [12] | DOI: 10.1007/978-94-011-2002-9 · doi:10.1007/978-94-011-2002-9 |
| [13] | DOI: 10.1007/BF01844169 · Zbl 0212.06401 · doi:10.1007/BF01844169 |
| [14] | DOI: 10.2307/1970044 · doi:10.2307/1970044 |
| [15] | Chen Y., Arabian J. Sci. Eng. 34 pp 1– |
| [16] | Cohn P. M., Proc. London Math. Soc. 11 pp 511– |
| [17] | DOI: 10.1090/S0002-9939-1952-0050566-9 · doi:10.1090/S0002-9939-1952-0050566-9 |
| [18] | Filippov V. T., Dokl. Akad. Nauk SSSR 260 pp 1082– |
| [19] | Filippov V. T., Trudy Inst. Mat., Nauka Sibirsk. Otdel., Novosibirsk 4 pp 139– |
| [20] | DOI: 10.1007/BF01105577 · Zbl 0309.20012 · doi:10.1007/BF01105577 |
| [21] | DOI: 10.1016/j.jpaa.2007.06.008 · Zbl 1185.16038 · doi:10.1016/j.jpaa.2007.06.008 |
| [22] | DOI: 10.1017/S1446788700018073 · Zbl 0296.20015 · doi:10.1017/S1446788700018073 |
| [23] | Higman G., J. London Math. Soc. 26 pp 61– |
| [24] | Higman G., J. London Math. Soc. 24 pp 247– |
| [25] | Howie J. M., Fundamentals of Semigroup Theory (1995) |
| [26] | Ivanov I. S., Trudy Mosk. Mat. Obshch. 17 pp 3– |
| [27] | Kurosh A. G., Siberian. Math. J. 1 pp 62– |
| [28] | Kurosh A. G., Russian Math. Surveys. Uspekhi 24 pp 3– |
| [29] | Lyndon R. C., Trans. Am. Math. Soc. 77 pp 202– |
| [30] | DOI: 10.1007/978-3-642-61896-3 · doi:10.1007/978-3-642-61896-3 |
| [31] | Malcev A. I., Uspekhi Mat. Nauk N.S. 7 pp 181– |
| [32] | Neumann B., Proc. London Math. Soc. 1 pp 241– |
| [33] | Rota G. C., Bull. Amer. Math. Soc. 5 pp 325– |
| [34] | DOI: 10.1007/BF01668599 · Zbl 0399.17006 · doi:10.1007/BF01668599 |
| [35] | Shirshov A. I., Mat. Sb. 45 pp 13– |
| [36] | Shirshov A. I., Mat. Sbornik 38 pp 149– |
| [37] | Skornyakov L. A., Mat. Sb. 44 pp 297– |
| [38] | Shutov E. G., Mat. Sb. 62 pp 496– |
| [39] | Ufnarovski V. A., Encyclopaedia Mat. Sci. 57 pp 1– · doi:10.1007/978-3-662-06292-0_1 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
