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Monoid varieties with extreme properties


Authors: Marcel Jackson and Edmond W. H. Lee
Journal: Trans. Amer. Math. Soc. 370 (2018), 4785-4812
MSC (2010): Primary 20M07
DOI: https://doi.org/10.1090/tran/7091
Published electronically: January 18, 2018
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Abstract: Finite monoids that generate monoid varieties with uncountably many subvarieties seem rare, and, surprisingly, no finite monoid is known to generate a monoid variety with countably infinitely many subvarieties. In the present article, it is shown that there are, nevertheless, many finite monoids with simple descriptions that generate monoid varieties with continuum many subvarieties; these include inherently nonfinitely based finite monoids and all monoids for which $ xyxy$ is an isoterm. It follows that the join of two Cross monoid varieties can have a continuum cardinality subvariety lattice that violates the ascending chain condition.

Regarding monoid varieties with countably infinitely many subvarieties, the first example of a finite monoid that generates such a variety is exhibited. A complete description of the subvariety lattice of this variety is given. This lattice has width three and contains only finitely based varieties, all except two of which are Cross.


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  • [1] Jorge Almeida, Finite semigroups and universal algebra, Translated from the 1992 Portuguese original and revised by the author. Series in Algebra, vol. 3, World Scientific Publishing Co., Inc., River Edge, NJ, 1994. MR 1331143
  • [2] A. P. Birjukov, Varieties of idempotent semigroups, Algebra i Logika 9 (1970), 255-273 (Russian). MR 0297897
  • [3] Stanley Burris and H. P. Sankappanavar, A course in universal algebra, Graduate Texts in Mathematics, vol. 78, Springer-Verlag, New York-Berlin, 1981. MR 648287
  • [4] Igor Dolinka, An uncountable interval in the lattice of semiring varieties, Acta Sci. Math. (Szeged) 74 (2008), no. 1-2, 85-93. MR 2431092
  • [5] Trevor Evans, The lattice of semigroup varieties, Semigroup Forum 2 (1971), no. 1, 1-43. MR 0284528, https://doi.org/10.1007/BF02572269
  • [6] Charles C. Edmunds, Edmond W. H. Lee, and Ken W. K. Lee, Small semigroups generating varieties with continuum many subvarieties, Order 27 (2010), no. 1, 83-100. MR 2601157, https://doi.org/10.1007/s11083-010-9142-8
  • [7] Charles F. Fennemore, All varieties of bands. I, II, Math. Nachr. 48 (1971), 237-252; ibid. 48 (1971), 253-262. MR 0294535, https://doi.org/10.1002/mana.19710480118
  • [8] J. A. Gerhard, The lattice of equational classes of idempotent semigroups, J. Algebra 15 (1970), 195-224. MR 0263953, https://doi.org/10.1016/0021-8693(70)90073-6
  • [9] T. J. Head, The varieties of commutative monoids, Nieuw Arch. Wisk. (3) 16 (1968), 203-206. MR 0237696
  • [10] Marcel Jackson, Finite semigroups whose varieties have uncountably many subvarieties, J. Algebra 228 (2000), no. 2, 512-535. MR 1764577, https://doi.org/10.1006/jabr.1999.8280
  • [11] Marcel Jackson, On the finite basis problem for finite Rees quotients of free monoids, Acta Sci. Math. (Szeged) 67 (2001), no. 1-2, 121-159. MR 1830137
  • [12] Marcel Jackson, Finite semigroups with infinite irredundant identity bases, Internat. J. Algebra Comput. 15 (2005), no. 3, 405-422. MR 2151419, https://doi.org/10.1142/S0218196705002335
  • [13] Marcel Jackson, Finiteness properties of varieties and the restriction to finite algebras, Semigroup Forum 70 (2005), no. 2, 159-187. MR 2129773, https://doi.org/10.1007/s00233-004-0161-x
  • [14] Marcel Jackson, Syntactic semigroups and the finite basis problem, Structural theory of automata, semigroups, and universal algebra, NATO Sci. Ser. II Math. Phys. Chem., vol. 207, Springer, Dordrecht, 2005, pp. 159-167. MR 2210129, https://doi.org/10.1007/1-4020-3817-8_6
  • [15] Marcel Jackson and Ralph McKenzie, Interpreting graph colorability in finite semigroups, Internat. J. Algebra Comput. 16 (2006), no. 1, 119-140. MR 2217645, https://doi.org/10.1142/S0218196706002846
  • [16] Marcel Jackson and Olga Sapir, Finitely based, finite sets of words, Internat. J. Algebra Comput. 10 (2000), no. 6, 683-708. MR 1809378, https://doi.org/10.1142/S0218196700000327
  • [17] J. Ježek, Intervals in the lattice of varieties, Algebra Universalis 6 (1976), no. 2, 147-158. MR 0419332, https://doi.org/10.1007/BF02485826
  • [18] Jiří Kaďourek, Uncountably many varieties of semigroups satisfying $ x^2y\bumpeq xy$, Semigroup Forum 60 (2000), no. 1, 135-152. MR 1829935, https://doi.org/10.1007/s002330010007
  • [19] Jiří Kaďourek, A finitely generated variety of combinatorial inverse semigroups having uncountably many subvarieties, Proc. Roy. Soc. Edinburgh Sect. A 132 (2002), no. 6, 1373-1394. MR 1950812
  • [20] Edmond W. H. Lee, Subvarieties of the variety generated by the five-element Brandt semigroup, Internat. J. Algebra Comput. 16 (2006), no. 2, 417-441. MR 2228521, https://doi.org/10.1142/S0218196706002998
  • [21] Edmond W. H. Lee, Minimal semigroups generating varieties with complex subvariety lattices, Internat. J. Algebra Comput. 17 (2007), no. 8, 1553-1572. MR 2378051, https://doi.org/10.1142/S0218196707004189
  • [22] Edmond W. H. Lee, Combinatorial Rees-Sushkevich varieties that are Cross, finitely generated, or small, Bull. Aust. Math. Soc. 81 (2010), no. 1, 64-84. MR 2584924, https://doi.org/10.1017/S0004972709000616
  • [23] Edmond W. H. Lee, Cross varieties of aperiodic monoids with central idempotents, Port. Math. 68 (2011), no. 4, 425-429. MR 2854133, https://doi.org/10.4171/PM/1900
  • [24] Edmond W. H. Lee, Varieties generated by 2-testable monoids, Studia Sci. Math. Hungar. 49 (2012), no. 3, 366-389. MR 3099233, https://doi.org/10.1556/SScMath.49.2012.3.1211
  • [25] Edmond W. H. Lee, Finite basis problem for semigroups of order five or less: generalization and revisitation, Studia Logica 101 (2013), no. 1, 95-115. MR 3019333, https://doi.org/10.1007/s11225-012-9369-z
  • [26] Edmond W. H. Lee, Inherently non-finitely generated varieties of aperiodic monoids with central idempotents, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 423 (2014), no. Voprosy Teorii Predstavleniĭ Algebr i Grupp. 26, 166-182; English transl., J. Math. Sci. (N.Y.) 209 (2015), no. 4, 588-599. MR 3480696
  • [27] Edmond W. H. Lee, On certain Cross varieties of aperiodic monoids with commuting idempotents, Results Math. 66 (2014), no. 3-4, 491-510. MR 3272641, https://doi.org/10.1007/s00025-014-0390-6
  • [28] Edmond W. H. Lee and Jian Rong Li, Minimal non-finitely based monoids, Dissertationes Math. (Rozprawy Mat.) 475 (2011), 65. MR 2663812, https://doi.org/10.4064/dm475-0-1
  • [29] Edmond W. H. Lee and Jian Rong Li, The variety generated by all monoids of order four is finitely based, Glas. Mat. Ser. III 50(70) (2015), no. 2, 373-396. MR 3437496, https://doi.org/10.3336/gm.50.2.08
  • [30] Edmond W. H. Lee, Jian Rong Li, and Wen Ting Zhang, Minimal non-finitely based semigroups, Semigroup Forum 85 (2012), no. 3, 577-580. MR 3001603, https://doi.org/10.1007/s00233-012-9434-y
  • [31] Edmond W. H. Lee and Mikhail V. Volkov, Limit varieties generated by completely 0-simple semigroups, Internat. J. Algebra Comput. 21 (2011), no. 1-2, 257-294. MR 2787461, https://doi.org/10.1142/S0218196711006169
  • [32] Edmond W. H. Lee and Wen Ting Zhang, The smallest monoid that generates a non-Cross variety, Xiamen Daxue Xuebao Ziran Kexue Ban 53 (2014), no. 1, 1-4 (Chinese, with English and Chinese summaries). MR 3222278
  • [33] Yanfeng Luo and Wenting Zhang, On the variety generated by all semigroups of order three, J. Algebra 334 (2011), 1-30. MR 2787650, https://doi.org/10.1016/j.jalgebra.2011.02.037
  • [34] A. I. Mal $ \textprime$cev, Algebraicheskie sistemy., Posthumous edition. Edited by D. Smirnov and M. Taĭclin, Izdat. ``Nauka'', Moscow, 1970 (Russian). MR 0282908
  • [35] Sheila Oates and M. B. Powell, Identical relations in finite groups, J. Algebra 1 (1964), 11-39. MR 0161904, https://doi.org/10.1016/0021-8693(64)90004-3
  • [36] Sheila Oates-MacDonald and M. R. Vaughan-Lee, Varieties that make one Cross, J. Austral. Math. Soc. Ser. A 26 (1978), no. 3, 368-382. MR 515754
  • [37] Sheila Oates-Williams, On the variety generated by Murskiĭ's algebra, Algebra Universalis 18 (1984), no. 2, 175-177. MR 743465, https://doi.org/10.1007/BF01198526
  • [38] Peter Perkins, Bases for equational theories of semigroups, J. Algebra 11 (1969), 298-314. MR 0233911, https://doi.org/10.1016/0021-8693(69)90058-1
  • [39] M. V. Sapir, Inherently non-finitely based finite semigroups, Mat. Sb. (N.S.) 133(175) (1987), no. 2, 154-166, 270 (Russian); English transl., Math. USSR-Sb. 61 (1988), no. 1, 155-166. MR 905002
  • [40] M. V. Sapir, Problems of Burnside type and the finite basis property in varieties of semigroups, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 2, 319-340, 447 (Russian); English transl., Math. USSR-Izv. 30 (1988), no. 2, 295-314. MR 897000
  • [41] Mark V. Sapir, Combinatorial algebra: syntax and semantics, With contributions by Victor S. Guba and Mikhail V. Volkov. Springer Monographs in Mathematics, Springer, Cham, 2014. MR 3243545
  • [42] Olga Sapir, Finitely based words, Internat. J. Algebra Comput. 10 (2000), no. 4, 457-480. MR 1776051, https://doi.org/10.1142/S0218196700000224
  • [43] Olga Sapir, Non-finitely based monoids, Semigroup Forum 90 (2015), no. 3, 557-586. MR 3345943, https://doi.org/10.1007/s00233-015-9708-2
  • [44] L. N. Shevrin, B. M. Vernikov, and M. V. Volkov, Lattices of semigroup varieties, Izv. Vyssh. Uchebn. Zaved. Mat. 3 (2009), 3-36 (Russian, with English and Russian summaries); English transl., Russian Math. (Iz. VUZ) 53 (2009), no. 3, 1-28. MR 2581451, https://doi.org/10.3103/S1066369X09030013
  • [45] D. V. Skokov and B. M. Vernikov, Chains and anti-chains in the lattice of epigroup varieties, Semigroup Forum 80 (2010), no. 2, 341-345. MR 2601769, https://doi.org/10.1007/s00233-009-9196-3
  • [46] A. N. Trakhtman, A six-element semigroup that generates a variety with a continuum of subvarieties, Ural. Gos. Univ. Mat. Zap. 14 (1988), no. 3, Algebr. Sistemy i ikh Mnogoobr., 138-143, v (Russian). MR 958333
  • [47] M. R. Vaughan-Lee, Uncountably many varieties of groups, Bull. London Math. Soc. 2 (1970), 280-286. MR 0276307, https://doi.org/10.1112/blms/2.3.280
  • [48] B. M. Vernikov and M. V. Volkov, Lattices of nilpotent varieties of semigroups. II, Izv. Ural. Gos. Univ. Mat. Mekh. 1(10) (1998), 13-33, 179 (Russian, with English and Russian summaries). MR 1784293
  • [49] M. V. Volkov, Reflexive relations, extensive transformations and piecewise testable languages of a given height, International Conference on Semigroups and Groups in honor of the 65th birthday of Prof. John Rhodes. Internat. J. Algebra Comput. 14 (2004), no. 5-6, 817-827. MR 2104784, https://doi.org/10.1142/S0218196704002018
  • [50] Wenting Zhang and Yanfeng Luo, The variety generated by a certain transformation monoid, Internat. J. Algebra Comput. 18 (2008), no. 7, 1193-1201. MR 2468743, https://doi.org/10.1142/S0218196708004810

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Additional Information

Marcel Jackson
Affiliation: Department of Mathematics and Statistics, La Trobe University, Victoria 3086, Australia
Email: m.g.jackson@latrobe.edu.au

Edmond W. H. Lee
Affiliation: Department of Mathematics, Nova Southeastern University, Fort Lauderale, Florida 33314
Email: edmond.lee@nova.edu

DOI: https://doi.org/10.1090/tran/7091
Received by editor(s): July 2, 2015
Received by editor(s) in revised form: September 20, 2016, and September 26, 2016
Published electronically: January 18, 2018
Additional Notes: The first author was supported by ARC Discovery Project DP1094578 and Future Fellowship FT120100666
Dedicated: Dedicated to the 81st birthday of John L. Rhodes
Article copyright: © Copyright 2018 American Mathematical Society

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