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Journal articles on the topic 'Shirshova'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Bezhenar, V. F., E. F. Kira, A. S. Turlak, B. N. Novikov, and K. E. Kira. "DMITRY IVANOVICH SHIRSHOV – DOCTOR, SCIENTIST, INNOVATOR (on the 150th anniversary of the birth)." Scientific Notes of the I. P. Pavlov St. Petersburg State Medical University 25, no. 4 (April 21, 2019): 15–20. http://dx.doi.org/10.24884/1607-4181-2018-25-4-15-20.

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IIn 2018, the 150th anniversary of Dmitrii I. Shirshov was celebrated. The article is devoted to Shirshov’s biography and describes his research pathway, the main achievements in obstetrics, gynecology and urogynecology. In addition, the article tells about the characteristic of his clinical and teaching activities.

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2

Bryant, R. M., L. G. Kovács, and Ralph Stöhr. "Subalgebras of free restricted Lie algebras." Bulletin of the Australian Mathematical Society 72, no. 1 (August 2005): 147–56. http://dx.doi.org/10.1017/s0004972700034936.

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A theorem independently due to A.I. Shirshov and E. Witt asserts that every subalgebra of a free Lie algebra (over a field) is free. The main step in Shirshov's proof is a little known but rather remarkable result: if a set of homogeneous elements in a free Lie algebra has the property that no element of it is contained in the subalgebra generated by the other elements, then this subset is a free generating set for the subalgebra it generates. Witt also proved that every subalgebra of a free restricted Lie algebra is free. Later G.P. Kukin gave a proof of this theorem in which he adapted Shirshov's argument. The main step is similar, but it has come to light that its proof contains substantial gaps. Here we give a corrected proof of this main step in order to justify its applications elsewhere.

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3

SHNEERSON, L. M. "IDENTITIES AND A BOUNDED HEIGHT CONDITION FOR SEMIGROUPS." International Journal of Algebra and Computation 13, no. 05 (October 2003): 565–83. http://dx.doi.org/10.1142/s0218196703001559.

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We consider two different types of bounded height condition for semigroups. The first one originates from the classical Shirshov's bounded height theorem for associative rings. The second which is weaker, in fact was introduced by Wolf and also used by Bass for calculating the growth of finitely generated (f.g.) nilpotent groups. Both conditions yield polynomial growth. We give the first two examples of f.g. semigroups which have bounded height and do not satisfy any nontrivial identity. One of these semigroups does not have bounded height in the sense of Shirshov and the other satisfies the classical bounded height condition. This develops further one of the main results of the author's paper (J. Algebra, 1993) where the first examples of f.g. semigroups of polynomial growth and without nontrivial identities were given.

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4

Qiu, Jianjun, and Yuqun Chen. "Gröbner–Shirshov bases for Lie Ω-algebras and free Rota–Baxter Lie algebras." Journal of Algebra and Its Applications 16, no. 10 (September 20, 2017): 1750190. http://dx.doi.org/10.1142/s0219498817501900.

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We generalize the Lyndon–Shirshov words to the Lyndon–Shirshov [Formula: see text]-words on a set [Formula: see text] and prove that the set of all the nonassociative Lyndon–Shirshov [Formula: see text]-words forms a linear basis of the free Lie [Formula: see text]-algebra on the set [Formula: see text]. From this, we establish Gröbner–Shirshov bases theory for Lie [Formula: see text]-algebras. As applications, we give Gröbner–Shirshov bases of a free [Formula: see text]-Rota–Baxter Lie algebra, of a free modified [Formula: see text]-Rota–Baxter Lie algebra, and of a free Nijenhuis Lie algebra and, then linear bases of these three algebras are obtained.

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5

Tunyaz, Rabigul, and Abdukadir Obul. "Gröbner–Shirshov pair of irreducible modules over quantized enveloping algebra of type An." Journal of Algebra and Its Applications 17, no. 02 (January 23, 2018): 1850035. http://dx.doi.org/10.1142/s0219498818500354.

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In this paper, first, we give a Gröbner–Shirshov pair of finite-dimensional irreducible module [Formula: see text] over [Formula: see text] the quantized enveloping algebra of type [Formula: see text] by using the double free module method and the known Gröbner–Shirshov basis of [Formula: see text] Then, by specializing a suitable version of [Formula: see text] at [Formula: see text] we get a Gröbner–Shirshov basis of [Formula: see text] and get a Gröbner–Shirshov pair for the finite-dimensional irreducible module [Formula: see text] over [Formula: see text].

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6

Lee, Dong-il. "Gröbner-Shirshov Bases for Exceptional Lie Superalgebras." Algebra Colloquium 22, no. 01 (January 7, 2015): 1–10. http://dx.doi.org/10.1142/s1005386715000024.

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7

Yilmaz, Erol, Cenap Özel, and Uğur Ustaoğlu. "Gröbner–Shirshov basis and reduced words for affine Weyl group Ãn." Journal of Algebra and Its Applications 13, no. 06 (April 20, 2014): 1450005. http://dx.doi.org/10.1142/s0219498814500054.

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Using Buchberger–Shirshov Algorithm, Composition–Diamond Lemma and partitions of integers we obtain the reduced Gröbner–Shirshov basis of Ãn and classify all reduced words of the affine Weyl group Ãn.

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8

Zhao, Xiangui, and Yang Zhang. "Gelfand-Kirillov Dimensions of Modules over Differential Difference Algebras." Algebra Colloquium 23, no. 04 (September 26, 2016): 701–20. http://dx.doi.org/10.1142/s1005386716000596.

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Differential difference algebras are generalizations of polynomial algebras, quantum planes, and Ore extensions of automorphism type and of derivation type. In this paper, we investigate the Gelfand-Kirillov dimension of a finitely generated module over a differential difference algebra through a computational method: Gröbner-Shirshov basis method. We develop the Gröbner-Shirshov basis theory of differential difference algebras, and of finitely generated modules over differential difference algebras, respectively. Then, via Gröbner-Shirshov bases, we give algorithms for computing the Gelfand-Kirillov dimensions of cyclic modules and finitely generated modules over differential difference algebras.

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9

Kalorkoti, K., and I. Stanciu. "Parametrized Gröbner–Shirshov bases." Communications in Algebra 45, no. 5 (October 7, 2016): 1996–2017. http://dx.doi.org/10.1080/00927872.2016.1226875.

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10

Bokut, L. A., Yuqun Chen, and Abdukadir Obul. "Some new results on Gröbner–Shirshov bases for Lie algebras and around." International Journal of Algebra and Computation 28, no. 08 (December 2018): 1403–23. http://dx.doi.org/10.1142/s0218196718400027.

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We review Gröbner–Shirshov bases for Lie algebras and survey some new results on Gröbner–Shirshov bases for [Formula: see text]-Lie algebras, Gelfand–Dorfman–Novikov algebras, Leibniz algebras, etc. Some applications are given, in particular, some characterizations of extensions of groups, associative algebras and Lie algebras are given.

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11

BOKUT, L. A., YUQUN CHEN, and CIHUA LIU. "GRÖBNER–SHIRSHOV BASES FOR DIALGEBRAS." International Journal of Algebra and Computation 20, no. 03 (May 2010): 391–415. http://dx.doi.org/10.1142/s0218196710005753.

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In this paper, we define the Gröbner–Shirshov basis for a dialgebra. The Composition–Diamond lemma for dialgebras is given then. As a result, we give Gröbner–Shirshov bases for the universal enveloping algebra of a Leibniz algebra, the bar extension of a dialgebra, the free product of two dialgebras, and Clifford dialgebra. We obtain some normal forms for algebras mentioned the above.

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12

Kubat, Łukasz, and Jan Okniński. "Gröbner-Shirshov Bases for Plactic Algebras." Algebra Colloquium 21, no. 04 (October 6, 2014): 591–96. http://dx.doi.org/10.1142/s1005386714000534.

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A finite Gröbner-Shirshov basis is constructed for the plactic algebra of rank 3 over a field K. It is also shown that plactic algebras of rank exceeding 3 do not have finite Gröbner-Shirshov bases associated to the natural degree-lexicographic ordering on the corresponding free algebra. The latter is in contrast with the case of a strongly related class of algebras, called Chinese algebras.

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13

Bokut, L. A., Yuqun Chen, and Zerui Zhang. "Gröbner–Shirshov bases method for Gelfand–Dorfman–Novikov algebras." Journal of Algebra and Its Applications 16, no. 01 (January 2017): 1750001. http://dx.doi.org/10.1142/s0219498817500013.

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We establish Gröbner–Shirshov base theory for Gelfand–Dorfman–Novikov algebras over a field of characteristic [Formula: see text]. As applications, a PBW type theorem in Shirshov form is given and we provide an algorithm for solving the word problem of Gelfand–Dorfman–Novikov algebras with finite homogeneous relations. We also construct a subalgebra of one generated free Gelfand–Dorfman–Novikov algebra which is not free.

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14

Bokut, L. A., Yuqun Chen, and Qiuhui Mo. "Gröbner–Shirshov bases for semirings." Journal of Algebra 385 (July 2013): 47–63. http://dx.doi.org/10.1016/j.jalgebra.2013.03.013.

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15

Lee, Dong-il, and Jeong-Yup Lee. "The structure of the full icosahedral group." Acta Crystallographica Section A Foundations and Advances 70, a1 (August 5, 2014): C527. http://dx.doi.org/10.1107/s2053273314094728.

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We consider the full icosahedral group, which is the Coxeter group of type H_3. The Coxeter groups appear naturally in geometry and algebra. In 1935, the finite Coxeter groups were classified by Coxeter in terms of Coxeter-Dynkin diagrams. We remark that the affine extensions of the Coxeter groups of types H are related to quasicrystals with tenfold symmetry. Our approach to understanding the structure of Coxeter groups is the noncommutative Groebner basis theory, which is called the Groebner-Shirshov basis theory. By completing the relations coming from a presentation of the Coxeter group, we find a Groebner-Shirshov basis to obtain a set of standard monomials. Especially, for the Coxeter group of type H_3, its Groebner-Shirshov basis and the corresponding standard monomials are constructed. Thus, we understand the algebra structure of the group algebra C[H_3], which is not commutative.

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16

BOKUT, L. A., YUQUN CHEN, and XIANGUI ZHAO. "GRÖBNER–SHIRSHOV BASES FOR FREE INVERSE SEMIGROUPS." International Journal of Algebra and Computation 19, no. 02 (March 2009): 129–43. http://dx.doi.org/10.1142/s0218196709005019.

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A new construction for free inverse semigroups was obtained by Poliakova and Schein in 2005. Based on their result, we find Gröbner–Shirshov bases for free inverse semigroups with respect to the deg-lex order of words. In particular, we give the (unique and shortest) normal forms in the classes of equivalent words of a free inverse semigroup together with the Gröbner–Shirshov algorithm to transform any word to its normal form.

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17

Cevik, Ahmet, and Amer Albargi. "A solution of the word problem for braid groups via the complex reflection group G12." Filomat 34, no. 2 (2020): 461–67. http://dx.doi.org/10.2298/fil2002461c.

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It is known that if there exists a Gr?bner-Shirshov basis for a group G, then we say that one of the decision problem, namely the word problem, is solvable for G as well. Therefore, as the main target of this paper, we will present a (non-commutative) Gr?bner-Shirshov basis for the braid group associated with the congruence classes of complex reflection group G12 which will give us normal forms of the elements of G12 and so will obtain a new algorithm to solve the word problem over it.

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18

GAO, XING, LI GUO, and SHANGHUA ZHENG. "CONSTRUCTION OF FREE COMMUTATIVE INTEGRO-DIFFERENTIAL ALGEBRAS BY THE METHOD OF GRÖBNER–SHIRSHOV BASES." Journal of Algebra and Its Applications 13, no. 05 (February 25, 2014): 1350160. http://dx.doi.org/10.1142/s0219498813501600.

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In this paper, we construct free commutative integro-differential algebras by applying the method of Gröbner–Shirshov bases. We establish the Composition-Diamond Lemma for free commutative differential Rota–Baxter (DRB) algebras of order n. We also obtain a weakly monomial order on these algebras, allowing us to obtain Gröbner–Shirshov bases for free commutative integro-differential algebras on a set. We finally generalize the concept of functional monomials to free differential algebras with arbitrary weight and generating sets from which to construct a canonical linear basis for free commutative integro-differential algebras.

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19

Huang, Juwei, and Yuqun Chen. "Gröbner–Shirshov Bases Theory for Trialgebras." Mathematics 9, no. 11 (May 26, 2021): 1207. http://dx.doi.org/10.3390/math9111207.

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We establish a method of Gröbner–Shirshov bases for trialgebras and show that there is a unique reduced Gröbner–Shirshov basis for every ideal of a free trialgebra. As applications, we give a method for the construction of normal forms of elements of an arbitrary trisemigroup, in particular, A.V. Zhuchok’s (2019) normal forms of the free commutative trisemigroups are rediscovered and some normal forms of the free abelian trisemigroups are first constructed. Moreover, the Gelfand–Kirillov dimension of finitely generated free commutative trialgebra and free abelian trialgebra are calculated, respectively.

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20

ALAHMADI, ADEL, HAMED ALSULAMI, S. K. JAIN, and EFIM ZELMANOV. "LEAVITT PATH ALGEBRAS OF FINITE GELFAND–KIRILLOV DIMENSION." Journal of Algebra and Its Applications 11, no. 06 (November 14, 2012): 1250225. http://dx.doi.org/10.1142/s0219498812502258.

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21

Li, Yu, Qiuhui Mo, and Xiangui Zhao. "Gröbner–Shirshov bases for brace algebras." Communications in Algebra 46, no. 11 (September 19, 2018): 4577–89. http://dx.doi.org/10.1080/00927872.2018.1448846.

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22

Chen, Yuqun, and Guangliang Zhang. "Gröbner–Shirshov bases for commutative dialgebras." Communications in Algebra 47, no. 4 (February 22, 2019): 1671–89. http://dx.doi.org/10.1080/00927872.2018.1513017.

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23

BOKUT, L. A., YUQUN CHEN, and JIAPENG HUANG. "GRÖBNER–SHIRSHOV BASES FOR L-ALGEBRAS." International Journal of Algebra and Computation 23, no. 03 (April 16, 2013): 547–71. http://dx.doi.org/10.1142/s0218196713500094.

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In this paper, we first establish Composition-Diamond lemma for Ω-algebras. We give a Gröbner–Shirshov basis of the free L-algebra as a quotient algebra of a free Ω-algebra, and then the normal form of the free L-algebra is obtained. Second we establish Composition-Diamond lemma for L-algebras. As applications, we give Gröbner–Shirshov bases of the free dialgebra and the free product of two L-algebras, and then we show four embedding theorems of L-algebras: (1) Every countably generated L-algebra can be embedded into a two-generated L-algebra. (2) Every L-algebra can be embedded into a simple L-algebra. (3) Every countably generated L-algebra over a countable field can be embedded into a simple two-generated L-algebra. (4) Three arbitrary L-algebras A, B, C over a field k can be embedded into a simple L-algebra generated by B and C if |k| ≤ dim (B * C) and |A| ≤ |B * C|, where B * C is the free product of B and C.

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24

Yunus, Gulshadam, Zhenzhen Gao, and Abdukadir Obul. "Gröbner-Shirshov Basis of Quantum Groups." Algebra Colloquium 22, no. 03 (July 14, 2015): 495–516. http://dx.doi.org/10.1142/s1005386715000449.

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In this paper, by using the Ringel-Hall algebra method, we prove that the set of the skew-commutator relations of quantum root vectors forms a minimal Gröbner-Shirshov basis for the quantum groups of Dynkin type. As an application, we give an explicit basis for the types E7 and Dn.

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25

Zhao, Xiangui. "Jacobson’s Lemma via Gröbner-Shirshov Bases." Algebra Colloquium 24, no. 02 (April 23, 2017): 309–14. http://dx.doi.org/10.1142/s1005386717000189.

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Let R be a ring with identity 1. Jacobson’s lemma states that for any [Formula: see text], if 1− ab is invertible then so is 1 − ba. Jacobson’s lemma has suitable analogues for several types of generalized inverses, e.g., Drazin inverse, generalized Drazin inverse, and inner inverse. In this note we give a constructive way via Gröbner-Shirshov basis theory to obtain the inverse of 1 − ab in terms of (1 − ba)−1, assuming the latter exists.

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26

Kolesnikov, P. S. "Gröbner–Shirshov Bases for Replicated Algebras." Algebra Colloquium 24, no. 04 (November 15, 2017): 563–76. http://dx.doi.org/10.1142/s1005386717000372.

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We establish a universal approach to solutions of the word problem in the varieties of di- and tri-algebras. This approach, for example, allows us to apply Gröbner–Shirshov bases method for Lie algebras to solve the ideal membership problem in free Leibniz algebras (Lie di-algebras). As another application, we prove an analogue of the Poincaré–Birkhoff–Witt Theorem for universal enveloping associative tri-algebra of a Lie tri-algebra (CTD!-algebra).

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27

Aldhous, P. "Shirshov Institute: Mixing Research and Commerce." Science 264, no. 5163 (May 27, 1994): 1278. http://dx.doi.org/10.1126/science.264.5163.1278.

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28

Bokut*, L. A., and L. S. Shiao. "GRÖBNER-SHIRSHOV BASES FOR COXETER GROUPS." Communications in Algebra 29, no. 9 (July 31, 2001): 4305–19. http://dx.doi.org/10.1081/agb-100106002.

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29

Kharchenko, V. K. "Shirshov finiteness over rings of constants." Algebra and Logic 36, no. 2 (March 1997): 133–44. http://dx.doi.org/10.1007/bf02672480.

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30

Ateş, Fırat, Eylem G. Karpuz, Canan Kocapınar, and A. Sinan Çevik. "Gröbner–Shirshov bases of some monoids." Discrete Mathematics 311, no. 12 (June 2011): 1064–71. http://dx.doi.org/10.1016/j.disc.2011.03.008.

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31

Bokut, L. A., and Yuqun Chen. "Gröbner–Shirshov bases and their calculation." Bulletin of Mathematical Sciences 4, no. 3 (September 9, 2014): 325–95. http://dx.doi.org/10.1007/s13373-014-0054-6.

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32

Petrov, Fedor, and Pasha Zusmanovich. "On Shirshov bases of graded algebras." Israel Journal of Mathematics 197, no. 1 (January 3, 2013): 23–28. http://dx.doi.org/10.1007/s11856-012-0175-0.

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33

Kharchenko, V. K. "Braided version of Shirshov–Witt theorem." Journal of Algebra 294, no. 1 (December 2005): 196–225. http://dx.doi.org/10.1016/j.jalgebra.2005.03.034.

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34

Lee, Dong-il. "Standard monomials for the Weyl group F4." Journal of Algebra and Its Applications 15, no. 08 (July 24, 2016): 1650146. http://dx.doi.org/10.1142/s0219498816501462.

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35

Li, Yu, and Qiuhui Mo. "On Free Pre-Lie Algebras." Algebra Colloquium 24, no. 02 (April 23, 2017): 267–84. http://dx.doi.org/10.1142/s1005386717000153.

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In this paper, we find Hall-Shirshov type bases for free pre-Lie algebras. We show that Segal’s basis of a free pre-Lie algebra is a type of these bases. We give a nonassociative Gröbner-Shirshov basis S for a free pre-Lie algebra such that Irr(S) is a monomial basis (called normal words) of a free pre-Lie algebra, where Irr(S) is the set of all nonassociative words, not containing maximal nonassociative words of polynomials from S. We establish the Composition-Diamond lemma for free pre-Lie algebras over the basis of normal words and the degree breadth lexicographic ordering.

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36

Kim, Sung Soon, and Dong-il Lee. "Gröbner–Shirshov bases for Temperley–Lieb algebras of types B and D." Journal of Algebra and Its Applications 19, no. 01 (January 29, 2019): 2050002. http://dx.doi.org/10.1142/s0219498820500024.

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37

Kocapinar, Canan, Eylem Güzel Karpuz, Firat Ateş, and A. Sinan Çevik. "Gröbner-Shirshov Bases of the Generalized Bruck-Reilly ∗-Extension." Algebra Colloquium 19, spec01 (October 31, 2012): 813–20. http://dx.doi.org/10.1142/s1005386712000703.

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CHEN, YUQUN, and JIANJUN QIU. "GRÖBNER–SHIRSHOV BASIS FOR THE CHINESE MONOID." Journal of Algebra and Its Applications 07, no. 05 (October 2008): 623–28. http://dx.doi.org/10.1142/s0219498808003028.

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Nigmatulin, R. I. "60 YEARS ANNIVERSARY OF ALEXEY V. SOKOV." XXII workshop of the Council of nonlinear dynamics of the Russian Academy of Sciences 47, no. 1 (April 30, 2019): 248–52. http://dx.doi.org/10.29006/1564-2291.jor-2019.47(1).53.

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After graduation with honors from the Department of Geography of Lomonosov Moscow State University in 1981, Alexey V. Sokov started working in the State Oceanographic Institute, where he held the positions of an engineer, researcher, senior researcher, and eventually deputy director for the scientific work. He worked there until 1993, and during the period of 1985-1989 he studied in Graduate school of the same institute and defended his PhD thesis in 1990. In 1993 he joined the Shirshov Institute of Oceanology as a senior research scientist. From 1996 to 1999 he was deputy director of the Institute for the scientific and organizational work, since 1999 deputy director for fleet. During his work at Shirshov Institute, Alexey Sokov participated in 32 scientific expeditions under the programs "Abyssal", "Meridian", as well as WOCE and CLIVAR international projects. He personally directed 22 of these expeditions. In 2012 he defended his Dr. Sci. dissertation and in 2016 was appointed as interim Director of the Shirshov Institute of Oceanology of the Russian Academy of Sciences. Currently is a member of the Scientific Expert Council of the Marine Board under the Government of the Russian Federation. He is the author of numerous scientific papers published in domestic and foreign scientific journals.

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40

Guo, Li, William Y. Sit, and Ronghua Zhang. "Differential type operators and Gröbner–Shirshov bases." Journal of Symbolic Computation 52 (May 2013): 97–123. http://dx.doi.org/10.1016/j.jsc.2012.05.014.

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41

Karpuz, Eylem Güzel, Firat Ateş, and A. Sinan Çevik. "Gröbner-Shirshov bases of some Weyl groups." Rocky Mountain Journal of Mathematics 45, no. 4 (August 2015): 1165–75. http://dx.doi.org/10.1216/rmj-2015-45-4-1165.

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42

Bokut, Leonid, and Andrei Vesnin. "Gröbner–Shirshov bases for some braid groups." Journal of Symbolic Computation 41, no. 3-4 (March 2006): 357–71. http://dx.doi.org/10.1016/j.jsc.2005.09.009.

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43

Yurkova, R. M., A. A. Peyve, V. P. Zinkevich, and V. I. Cherkashin. "AMPHIBOLITES OF THE SHIRSHOV RIDGE, BERING SEA." International Geology Review 27, no. 9 (September 1985): 1051–68. http://dx.doi.org/10.1080/00206818509466482.

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44

Kolesnikov, Pavel. "Gröbner–Shirshov bases for pre-associative algebras." Communications in Algebra 45, no. 12 (March 15, 2017): 5283–96. http://dx.doi.org/10.1080/00927872.2017.1304552.

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45

BOKUT, L. A., YUQUN CHEN, and QIUHUI MO. "GRÖBNER–SHIRSHOV BASES AND EMBEDDINGS OF ALGEBRAS." International Journal of Algebra and Computation 20, no. 07 (November 2010): 875–900. http://dx.doi.org/10.1142/s0218196710005923.

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In this paper, 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 Ω-algebras, associative λ-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 Ω-algebras, associative λ-differential algebras. We give another 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).

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46

Lee, Dong-Il. "GROBNER-SHIRSHOV BASES FOR IRREDUCIBLE sp4-MODULES." Journal of the Korean Mathematical Society 45, no. 3 (May 31, 2008): 711–25. http://dx.doi.org/10.4134/jkms.2008.45.3.711.

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47

Chen, Yuqun. "Gröbner–Shirshov Bases for Extensions of Algebras." Algebra Colloquium 16, no. 02 (June 2009): 283–92. http://dx.doi.org/10.1142/s1005386709000285.

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An algebra [Formula: see text] is called an extension of the algebra M by B if M2 = 0, M is an ideal of [Formula: see text] and [Formula: see text] as algebras. In this paper, by using Gröbner–Shirshov bases, we characterize completely the extensions of M by B. An algorithm to find the conditions of an algebra A to be an extension of M by B is obtained.

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48

Karpuz, Eylem Güzel. "Gröbner-Shirshov Bases of Some Semigroup Constructions." Algebra Colloquium 22, no. 01 (January 7, 2015): 35–46. http://dx.doi.org/10.1142/s100538671500005x.

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49

Chen, Yu, Y. Li, and Q. Tang. "Gröbner–Shirshov bases for some Lie algebras." Siberian Mathematical Journal 58, no. 1 (January 2017): 176–82. http://dx.doi.org/10.1134/s0037446617010220.

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50

Kelarev, Andrei, John Yearwood, and Paul Watters. "INTERNET SECURITY APPLICATIONS OF GRÖBNER-SHIRSHOV BASES." Asian-European Journal of Mathematics 03, no. 03 (September 2010): 435–42. http://dx.doi.org/10.1142/s1793557110000283.

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This article is motivated by internet security applications of multiple classifiers designed for the detection of malware. Following a standard approach in data mining, Dazeley et al. (Asian-European J. Math. 2 (2009)(1) 41–56) used Gröbner-Shirshov bases to define a family of multiple classifiers and develop an algorithm optimizing their properties.The present article complements and strengthens these results. We consider a broader construction of classifiers and develop a new and more general algorithm for the optimization of their essential properties.

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