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Maximal subgroups of free idempotent generated semigroups over the full linear monoid. (English) Zbl 1297.20059

The authors show that the rank \(r\) component of the free idempotent generated semigroup of the biordered set of the full linear monoid of \(n\times n\) matrices over a division ring \(Q\) has a maximal subgroup isomorphic to the general linear group \(\text{GL}_r(Q)\), where \(n\) and \(r\) are positive integers with \(r<n/3\).

MSC:

20M05 Free semigroups, generators and relations, word problems
20M32 Algebraic monoids
20G15 Linear algebraic groups over arbitrary fields
15A30 Algebraic systems of matrices
57M15 Relations of low-dimensional topology with graph theory
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[1] Jorge Almeida, Stuart Margolis, Benjamin Steinberg, and Mikhail Volkov, Representation theory of finite semigroups, semigroup radicals and formal language theory, Trans. Amer. Math. Soc. 361 (2009), no. 3, 1429 – 1461. · Zbl 1185.20058
[2] Mark Brittenham, Stuart W. Margolis, and John Meakin, Subgroups of the free idempotent generated semigroups need not be free, J. Algebra 321 (2009), no. 10, 3026 – 3042. · Zbl 1177.20064 · doi:10.1016/j.jalgebra.2008.12.017
[3] M. Brittenham, S. W. Margolis, and J. Meakin, Subgroups of free idempotent generated semigroups: full linear monoids. arXiv: 1009.5683. · Zbl 1177.20064
[4] Peter J. Cameron and Csaba Szabó, Independence algebras, J. London Math. Soc. (2) 61 (2000), no. 2, 321 – 334. · Zbl 0968.08002 · doi:10.1112/S0024610799008546
[5] A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. I, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1961. · Zbl 0111.03403
[6] Igor Dolinka, A note on maximal subgroups of free idempotent generated semigroups over bands, Period. Math. Hungar. 65 (2012), no. 1, 97 – 105. · Zbl 1269.20050 · doi:10.1007/s10998-012-2776-0
[7] D. Easdown, Biordered sets come from semigroups, J. Algebra 96 (1985), no. 2, 581 – 591. · Zbl 0602.20055 · doi:10.1016/0021-8693(85)90028-6
[8] D. Easdown, M. V. Sapir, and M. V. Volkov, Periodic elements of the free idempotent generated semigroup on a biordered set, Internat. J. Algebra Comput. 20 (2010), no. 2, 189 – 194. · Zbl 1200.20041 · doi:10.1142/S0218196710005583
[9] J. A. Erdos, On products of idempotent matrices, Glasgow Math. J. 8 (1967), 118 – 122. · Zbl 0157.07101 · doi:10.1017/S0017089500000173
[10] D. G. Fitz-Gerald, On inverses of products of idempotents in regular semigroups, J. Austral. Math. Soc. 13 (1972), 335 – 337. · Zbl 0244.20079
[11] V. Gould, Independence algebras, Algebra Universalis 33 (1995), no. 3, 294 – 318. · Zbl 0827.20075 · doi:10.1007/BF01190702
[12] R. L. Graham, On finite 0-simple semigroups and graph theory, Math. Systems Theory 2 (1968), 325 – 339. · Zbl 0177.03103 · doi:10.1007/BF01703263
[13] N. Graham, R. Graham, and J. Rhodes, Maximal subsemigroups of finite semigroups, J. Combinatorial Theory 4 (1968), 203 – 209. · Zbl 0157.04901
[14] R. Gray and N. Ruškuc, Generating sets of completely 0-simple semigroups, Comm. Algebra 33 (2005), no. 12, 4657 – 4678. · Zbl 1102.20038 · doi:10.1080/00927870500276676
[15] R. Gray and N. Ruskuc, On maximal subgroups of free idempotent generated semigroups, Israel J. Math. 189 (2012), 147 – 176. · Zbl 1276.20063 · doi:10.1007/s11856-011-0154-x
[16] R. Gray and N. Ruškuc, Maximal subgroups of free idempotent-generated semigroups over the full transformation monoid, Proc. Lond. Math. Soc. (3) 104 (2012), no. 5, 997 – 1018. · Zbl 1254.20054 · doi:10.1112/plms/pdr054
[17] J. A. Green, On the structure of semigroups, Ann. of Math. (2) 54 (1951), 163 – 172. · Zbl 0043.25601 · doi:10.2307/1969317
[18] Peter M. Higgins, Techniques of semigroup theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992. With a foreword by G. B. Preston. · Zbl 0744.20046
[19] C. H. Houghton, Completely 0-simple semigroups and their associated graphs and groups, Semigroup Forum 14 (1977), no. 1, 41 – 67. · Zbl 0358.20071 · doi:10.1007/BF02194654
[20] J. M. Howie, The subsemigroup generated by the idempotents of a full transformation semigroup, J. London Math. Soc. 41 (1966), 707 – 716. · Zbl 0146.02903 · doi:10.1112/jlms/s1-41.1.707
[21] J. M. Howie, Idempotents in completely 0-simple semigroups, Glasgow Math. J. 19 (1978), no. 2, 109 – 113. · Zbl 0393.20044 · doi:10.1017/S0017089500003475
[22] John M. Howie, Fundamentals of semigroup theory, London Mathematical Society Monographs. New Series, vol. 12, The Clarendon Press, Oxford University Press, New York, 1995. Oxford Science Publications. · Zbl 0835.20077
[23] Thomas J. Laffey, Products of idempotent matrices, Linear and Multilinear Algebra 14 (1983), no. 4, 309 – 314. · Zbl 0526.15008 · doi:10.1080/03081088308817567
[24] Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory, Second revised edition, Dover Publications, Inc., New York, 1976. Presentations of groups in terms of generators and relations. · Zbl 0362.20023
[25] Brett McElwee, Subgroups of the free semigroup on a biordered set in which principal ideals are singletons, Comm. Algebra 30 (2002), no. 11, 5513 – 5519. · Zbl 1017.20048 · doi:10.1081/AGB-120015667
[26] K. S. S. Nambooripad, Structure of regular semigroups. I, Mem. Amer. Math. Soc. 22 (1979), no. 224, vii+119. · Zbl 0457.20051 · doi:10.1090/memo/0224
[27] K. S. S. Nambooripad and F. Pastijn, Subgroups of free idempotent generated regular semigroups, Semigroup Forum 21 (1980), no. 1, 1 – 7. · Zbl 0449.20061 · doi:10.1007/BF02572533
[28] Jan Okniński, Semigroups of matrices, Series in Algebra, vol. 6, World Scientific Publishing Co., Inc., River Edge, NJ, 1998. · Zbl 0911.20042
[29] Jan Okniński and Mohan S. Putcha, Complex representations of matrix semigroups, Trans. Amer. Math. Soc. 323 (1991), no. 2, 563 – 581. · Zbl 0745.20057
[30] Mohan S. Putcha, Linear algebraic monoids, London Mathematical Society Lecture Note Series, vol. 133, Cambridge University Press, Cambridge, 1988. · Zbl 0647.20066
[31] Mohan S. Putcha, Monoids on groups with \?\?-pairs, J. Algebra 120 (1989), no. 1, 139 – 169. · Zbl 0683.20051 · doi:10.1016/0021-8693(89)90193-2
[32] Mohan S. Putcha, Classification of monoids of Lie type, J. Algebra 163 (1994), no. 3, 636 – 662. · Zbl 0801.20051 · doi:10.1006/jabr.1994.1035
[33] Mohan S. Putcha, Monoids of Lie type, Semigroups, formal languages and groups (York, 1993) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 466, Kluwer Acad. Publ., Dordrecht, 1995, pp. 353 – 367. · Zbl 0870.20046
[34] Mohan S. Putcha, Products of idempotents in algebraic monoids, J. Aust. Math. Soc. 80 (2006), no. 2, 193 – 203. · Zbl 1102.20043 · doi:10.1017/S1446788700013070
[35] Lex E. Renner, Linear algebraic monoids, Encyclopaedia of Mathematical Sciences, vol. 134, Springer-Verlag, Berlin, 2005. Invariant Theory and Algebraic Transformation Groups, V. · Zbl 1085.20041
[36] N. Ruškuc, Presentations for subgroups of monoids, J. Algebra 220 (1999), no. 1, 365 – 380. · Zbl 0943.20057 · doi:10.1006/jabr.1999.7930
[37] Louis Solomon, An introduction to reductive monoids, Semigroups, formal languages and groups (York, 1993) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 466, Kluwer Acad. Publ., Dordrecht, 1995, pp. 295 – 352. · Zbl 0870.20047
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