×

Recurrence relations and Hilbert series of the monoid associated with star topology. (English) Zbl 1489.20024

Summary: Affine monoids are the considered as natural discrete analogues of the finitely generated cones. The interconnection between these two objects has been an active area of research since last decade. Star network is one of the most common in computer network topologies. In this work, we study star topology \(S_n\) and associate a Coxeter structure of affine type on it. We find a recurrence relation and the Hilbert series of the associated right-angled monoid \(M(S_n^\infty)\). We observe that the growth rate of the monoid \(M(S_n^\infty)\) is unbounded.

MSC:

20M05 Free semigroups, generators and relations, word problems
20F36 Braid groups; Artin groups
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Stanley, R. P., Hilbert functions of graded algebras, Advances in Mathematics, 28, 1, 57-83 (1978) · Zbl 0384.13012 · doi:10.1016/0001-8708(78)90045-2
[2] Hal, S., Computational Algebraic Geometry (2003), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 1046.14034
[3] Bourbaki, N., Groupes et algèbres de Lie, Chapitres 4-6 (1968), Hermann, MO, USA: Elementary Mathematics, Hermann, MO, USA · Zbl 0186.33001
[4] Saito, K., Growth functions for Artin monoids, Proceedings of the Japan Academy, Series A, Mathematical Sciences, 85, 7, 84-88 (2009) · Zbl 1183.20061 · doi:10.3792/pjaa.85.84
[5] Artin, E., Theory of braids, The Annals of Mathematics, 48, 1, 101-126 (1947) · Zbl 0030.17703 · doi:10.2307/1969218
[6] Iqbal, Z., Hilbert series of positive braids, Algebra Colloquium, 18, 1017-1028 (2011) · Zbl 1297.20060 · doi:10.1142/s1005386711000897
[7] Iqbal, Z.; Yousaf, S., Hilbert series of the braid monoid \(MB_4\) in band generators, Turkish Journal of Mathematics, 38, 977-984 (2014) · Zbl 1310.20048 · doi:10.3906/mat-1401-58
[8] Mairesse, J.; Mathéus, F., Growth series for Artin groups of dihedral type, International Journal of Algebra and Computation, 16, 6, 1087-1107 (2006) · Zbl 1142.20018 · doi:10.1142/s0218196706003360
[9] Parry, W., Growth series of Coxeter groups and salem numbers, Journal of Algebra, 154, 2, 406-415 (1993) · Zbl 0796.20031 · doi:10.1006/jabr.1993.1022
[10] Berceanu, B.; Iqbal, Z., Universal upper bound for the growth of Artin monoids, Communications in Algebra, 43, 5, 1967-1982 (2015) · Zbl 1322.20048 · doi:10.1080/00927872.2014.881834
[11] Iqbal, Z.; Batool, S.; Akram, M., Hilbert series of right-angled affine Artin monoid \(M\left( \widetilde{A}_n^\infty\right)\), Kuwait Journal of Science, 44, 4, 19-27 (2017) · Zbl 1463.20061
[12] Young, C.; Iqbal, Z.; Rauf Nizami, A.; Munir, M.; Riaz, S.; Shin, M., Some recurrence relations and Hilbert series of right-angled affine Artin monoid \(M\left( \widetilde{D}_n^\infty\right)\), Journal of Function Spaces, 2018 (2018) · Zbl 1401.20065 · doi:10.1155/2018/1901657
[13] Coxeter, H. S. M., Regular Complex Polytopes (1991), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0732.51002
[14] Harpe, P. D., Topics in Geometric Group Theory (2000), Chicago, IL, USA: The University of Chicago Press, Chicago, IL, USA · Zbl 0965.20025
[15] Anick, D. J., On the homology of associative algebras, Transactions of the American Mathematical Society, 296, 2, 641 (1986) · Zbl 0598.16028 · doi:10.1090/s0002-9947-1986-0846601-5
[16] Bergman, G. M., The diamond lemma for ring theory, Advances in Mathematics, 29, 2, 178-218 (1978) · Zbl 0326.16019 · doi:10.1016/0001-8708(78)90010-5
[17] Bokut, L. A.; Fong, Y.; Ke, W.-F.; Shiao, L.-S., Gröbner-Shirshov bases for braid semigroup, Advances in Algebra, 60-72 (2003), Singapore: World Scientific Publishing, Singapore · Zbl 1053.20049
[18] Brown, K. S.; Baumslag, G.; Miller, C. F., The geometry of rewriting systems:a proof of Anick-Groves-Squeir theorem, Algorithms and Classification in Combinatorial Group Theory, 137-164 (1992), New York, NY, USA: Springer, New York, NY, USA
[19] Cohn, P. M., Further Algebra and Applications (2003), London, UK: Springer, London, UK · Zbl 1006.00001
[20] Ufnarovskij, V. A., Combinatorial and asymptotic methods in algebra, Encyclopaedia of Mathematical Sciences (1995), Berlin, Germany: Springer, Berlin, Germany · Zbl 0826.16001
[21] Kelley, W. G.; Peterson, A. C., Difference equations: An Introduction with Applications,, 125 (2001), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0970.39001
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.