A Gröbner-Shirshov basis over a special type of braid monoids

Ahmet S. Cevik; Eylem G. Karpuz; Hamed H. Alsulami; Esra K. Cetinalp; Ahmet S. Cevik; Eylem G. Karpuz; Hamed H. Alsulami; Esra K. Cetinalp

doi:10.3934/math.2020278

AIMS Mathematics

2020, Volume 5, Issue 5: 4357-4370. doi: 10.3934/math.2020278

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Research article

A Gröbner-Shirshov basis over a special type of braid monoids

1.
Department of Mathematics, KAU King Abdulaziz University, Science Faculty, 21589, Jeddah-Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Selcuk University, 42075, Konya-Turkey
3.
Department of Mathematics, Kamil Özdag Science Faculty, Karamanoglu Mehmetbey University, Yunus Emre Campus, 70100, Karaman-Turkey

Received: 19 March 2020 Accepted: 05 May 2020 Published: 11 May 2020
MSC : 13P10, 16S15, 20M05

The aim of this paper is to present a Gröbner-Shirshov basis for a special type of braid monoids, namely the symmetric inverse monoid I_n, in terms of the dex-leg ordering on the related elements of monoid. By taking into account the Gröbner-Shirshov basis, the ideal form (or, equivalently, the normal form structure) of this important monoid will be obtained. This ideal form will give us the solution of the word problem. At the final part of this paper, we give an application of our main result which find out a Gröbner-Shirshov basis for the symmetric inverse monoid I₄ such that the accuracy and efficiency of this example can be seen by GBNP package in GAP (Group, Algorithms and Programming) which computes Gröbner bases of non-commutative polynomials ^[1].
- Gröbner-Shirshov basis,
- symmetric inverse monoid,
- normal form,
- word problem,
- algorithm
Citation: Ahmet S. Cevik, Eylem G. Karpuz, Hamed H. Alsulami, Esra K. Cetinalp. A Gröbner-Shirshov basis over a special type of braid monoids[J]. AIMS Mathematics, 2020, 5(5): 4357-4370. doi: 10.3934/math.2020278

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Abstract

The aim of this paper is to present a Gröbner-Shirshov basis for a special type of braid monoids, namely the symmetric inverse monoid I_n, in terms of the dex-leg ordering on the related elements of monoid. By taking into account the Gröbner-Shirshov basis, the ideal form (or, equivalently, the normal form structure) of this important monoid will be obtained. This ideal form will give us the solution of the word problem. At the final part of this paper, we give an application of our main result which find out a Gröbner-Shirshov basis for the symmetric inverse monoid I₄ such that the accuracy and efficiency of this example can be seen by GBNP package in GAP (Group, Algorithms and Programming) which computes Gröbner bases of non-commutative polynomials ^[1].

References

[1]	A. M. Cohen, J. W. Knopper, Computing Gröbner bases of noncommutative polynomials, GBNP (version 1.0.3), 2016. Available from: https://www.gap-system.org/Packages/gbnp.html.
[2]	B. Buchberger, An algorithm for finding a basis for the residue class ring of a zero-dimensional polynomial ideal (in German). PhD thesis, University of Innsbruck, Innsbruck, Austria, 1965.
[3]	G. M. Bergman, The diamond lemma for ring theory, Adv. Math., 29 (1978), 178-218. doi: 10.1016/0001-8708(78)90010-5
[4]	A. I. Shirshov, Certain algorithmic problem for Lie algebras (in Russian), Sibirskii Math. Z., 3 (1962), 292-296; English translation in SIGSAM Bull., 33 (1999), 3-6.
[5]	L. A. Bokut, Embeddings into simple associative algebras, Algebr. Log., 15 (1976), 73-90. doi: 10.1007/BF01877233
[6]	F. Ates, E. G. Karpuz, C. Kocapinar, et al. Gröbner-Shirshov bases of some monoids, Discrete Math., 311 (2011), 1064-1071. doi: 10.1016/j.disc.2011.03.008
[7]	F. Ates, E. G. Karpuz, C. Kocapinar, et al. Gröbner-Shirshov basis for the singular part of the Brauer semigroup, Turk. J. Math., 42 (2018), 1338-1347.
[8]	L. A. Bokut, Gröbner-Shirshov basis for the Braid group in the Birman-Ko-Lee generators, J. Algebr., 321 (2009), 361-376. doi: 10.1016/j.jalgebra.2008.10.007
[9]	L. A. Bokut, Y. Chen, X. Zhao, Gröbner-Shirshov bases for free inverse semigroups, Int. J. Algebr. Comput., 19 (2009), 129-143. doi: 10.1142/S0218196709005019
[10]	E. G. Karpuz, Gröbner-Shirshov bases of some semigroup constructions, Algebr. Colloq., 22 (2015), 35-46. doi: 10.1142/S100538671500005X
[11]	E. G. Karpuz, F. Ates, A. S. Cevik, Gröbner-Shirshov bases of some Weyl groups, Rocky MT. J. Math., 45 (2015), 1165-1175. doi: 10.1216/RMJ-2015-45-4-1165
[12]	C. Kocapinar, E. G. Karpuz, F. Ates, et al. Gröbner-Shirshov bases of the generalized Bruck-Reilly *-extension, Algeb. Colloq., 19 (2012), 813-820. doi: 10.1142/S1005386712000703
[13]	A. I. Shirshov, Selected works of A.I.Shirshov, Birkhauser Basel, 2009.
[14]	Y. Chen, B. Wang, Gröbner-Shirshov bases and Hilbert series of free dendriform algebras, Southeast Asian Bull. Math., 34 (2010), 639-650.
[15]	Z. Iqbal, S. Yousaf, Hilbert series of the braid monoid MB4 in band generators, Turk. J. Math., 38 (2014), 977-984. doi: 10.3906/mat-1401-58
[16]	E. G. Karpuz, F. Ates, A. S. Cevik, et al. The graph based on Gröbner-Shirshov bases of groups, Fixed Point Theory Appl., 71 (2013).
[17]	U. Ali, B. Berceanu, Canonical forms of positive braids, J. Algebra Appl., 14 (2015), 1450076.
[18]	S. I. Adian, V. G. Durnev, Decision problems for groups and semigroups, Russ. Math. Surv., 55 (2000), 207-296. doi: 10.1070/RM2000v055n02ABEH000267
[19]	F. L. Pritchard, The ideal membership problem in non-commutative polynomial rings, J. Symb. Comput., 22 (1996), 27-48. doi: 10.1006/jsco.1996.0040
[20]	D. Easdown, T. G. Lavers, The inverse braid monoid, Adv. Math., 186 (2004), 438-455. doi: 10.1016/j.aim.2003.07.014
[21]	E. H. Moore, Concerning the abstract groups of order k! and $\frac{1}{2}k!$ holohedrically isomorphic with the symmetric and alternating substitution groups on k letters, Proc. London Math. Soc., 28 (1897), 357-366.
[22]	L. M. Popova, Defining relations in some semigroups of partial transformations of a finite set (in Russian), Uchenye Zap. Leningrad. Gos. Ped. Inst., 218 (1961), 191-212.
[23]	J. East, A symmetrical presentation for the singular part of the symmetric inverse monoid, Algebr. Univ., 74 (2015), 207-228. doi: 10.1007/s00012-015-0347-y
[24]	L. A. Bokut, Unsolvability of the word problem, and subalgebras of finitely presented Lie algebras, Izv. Akad. Nauk SSSR Math., 36 (1972), 1173-1219.
[25]	B. Buchberger, An algorithmic criteria for the solvability of algebraic systems of equations (in German), Aequationes Math., 4 (1970), 374-383. doi: 10.1007/BF01844169
[26]	L. A. Bokut, Y. Chen, Gröbner-Shirshov bases and their calculation, Bull. Math. Sci., 4 (2013), 325-395.

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