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Special elements in lattices of semigroup varietie

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Abstract

We survey results concerning special elements of eight types (modular, sublattices, namely, the lattices of commutative varieties, of permutative three of its sublattices, namely, the lattices of commutative varieties, of Ježek, McKenzie, Shaprynskiĭ Volkov and the author. Several open questions are formulated.

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References

  1. S. Burris and E. Nelson, Embedding the dual of Π∞ in the lattice of equational classes of semigroups, Algebra Universalis, 1 (1971), 248–254.

    MathSciNet  MATH  Google Scholar 

  2. S. Burris and E. Nelson, Embedding the dual of Πm in the lattice of equational classes of commutative semigroups, Proc. Amer. Math. Soc., 30 (1971), 37–39.

    MathSciNet  MATH  Google Scholar 

  3. J. A. Gerhard, Semigroups with an idempotent power. II. The lattice of equational subclasses of [(xy)2 = xy], Semigroup Forum, 14 (1977), 375–388.

    MathSciNet  MATH  Google Scholar 

  4. G. Grätzer, Lattice Theory: Foundation, Springer Basel AG, 2011.

    MATH  Google Scholar 

  5. G. Grätzer and E. T. Schmidt, Standard ideals in lattices, Acta Math. Acad. Sci. Hungar., 12 (2011), 17–86.

    MathSciNet  MATH  Google Scholar 

  6. J. Ježek, Intervals in lattices of varieties, Algebra Universalis, 6 (1976), 147–158.

    MathSciNet  MATH  Google Scholar 

  7. J. Ježek, The lattice of equational theories. Part I: modular elements, Czechosl. Math. J., 31 (1981), 127–152.

    MATH  Google Scholar 

  8. J. Ježek and R. N. McKenzie, Definability in the lattice of equational theories of semigroups, Semigroup Forum, 46 (1993), 199–245.

    MathSciNet  MATH  Google Scholar 

  9. A. Kisielewicz, Varieties of commutative semigroups, Trans. Amer. Math. Soc., 342 (1994), 275–306.

    MathSciNet  MATH  Google Scholar 

  10. F. J. Pastijn and P. G. Trotter, Complete congruences on lattices of varieties and of pseudovarieties, Int. J. Algebra and Comput., 8 (1998), 171–201.

    MathSciNet  MATH  Google Scholar 

  11. P. Perkins, Bases for equational theories of semigroups, J. Algebra, 11 (1969), 298–314.

    MathSciNet  MATH  Google Scholar 

  12. V. V. Rasin, Varieties of orthodox cliffordian semigroups, Izvestiya VUZ. Matematika (1982), no. 11, 82–85 [in Russian

    MathSciNet  MATH  Google Scholar 

  13. V. V. Rasin, Engl. translation: Soviet Math. Izv. VUZ, 26:11 (1982), 107–110.

    Google Scholar 

  14. M. V. Sapir and E. V. Sukhanov, On varieties of periodic semigroups, Izvestiya VUZ. Matematika (1981), no. 4, 48–55 [in Russian

    MathSciNet  MATH  Google Scholar 

  15. M. V. Sapir and E. V. Sukhanov Engl. translation: Soviet Math. Izv. VUZ, 25:4 (1981), 53–63].

    Google Scholar 

  16. V. Yu. Shaprynskiĭ, Distributive and neutral elements of the lattice of commutative semigroup varieties, Izvestiya VUZ. Matematika (2011), no. 7, 67–79 [Russian

    MathSciNet  MATH  Google Scholar 

  17. V. Yu. Shaprynskiĭ Engl. translation: Russ. Math. Izv. VUZ, 55:7 (2011), 56–67].

    MathSciNet  Google Scholar 

  18. V. Yu. Shaprynskiĭ, Modular and lower-modular elements of lattices of semigroup varieties, Semigroup Forum, 85 (2012), 97–110.

    MathSciNet  MATH  Google Scholar 

  19. V. Yu. Shaprynskiĭ, Periodicity of special elements of the lattice of semigroup varieties, Proc. Institute of Math. and Mechan. of the Ural Branch of the Russ. Acad. Sci., 18:3 (2012), 282–286 [in Russian].

    Google Scholar 

  20. V.Yu. Shaprynskiĭ, Identities and quasiidentities in the lattice of overcommutative semigroup varieties, Semigroup Forum, 86 (2013), 202–211.

    MathSciNet  MATH  Google Scholar 

  21. V. Yu. Shaprynskiĭ and B. M. Vernikov, Lower-modular elements of the lattice of semigroup varieties. III, Acta Sci. Math. (Szeged), 76 (2010), 371–382.

    MathSciNet  MATH  Google Scholar 

  22. V. Yu. Shaprynskiĭ and B. M. Vernikov, Special elements in the lattice of overcommutative semigroup varieties revisited, Order, 28 (2011), 139–155.

    MathSciNet  MATH  Google Scholar 

  23. L. N. Shevrin, B. M. Vernikov and M. V. Volkov, Lattices of semigroup varieties, Izvestiya VUZ. Matematika, (2009), 3–36. [Russian

    Google Scholar 

  24. L. N. Shevrin, B. M. Vernikov and M. V. Volkov Engl. translation: Russ. Math. Izv. VUZ, 53:3 (2009), 1–28].

  25. A. N. Trakhtman, On covering elements in the lattice of varieties of algebras, Matematicheskie Zametki, 15 (1974), 307–312 (in Russian)

    MathSciNet  Google Scholar 

  26. A. N. Trakhtman Engl. translation: Math. Notes, 15 (1974), 174–177.

    Google Scholar 

  27. B.M. Vernikov, Special elements in the lattice of overcommutative semigroup varieties, Matematicheskie Zametki, 70 (2001), 670–678 (in Russian); Engl. translation: Math. Notes, 70 (2001), 608–615; Correct. ibid., 88 (2010), 953–954 (in Russian)

    Google Scholar 

  28. B.M. Vernikov Engl. translation: Math. Notes, 88 (2010), 908–909.

    MathSciNet  Google Scholar 

  29. B. M. Vernikov, On modular elements of the lattice of semigroup varieties, Comment. Math. Univ. Carol., 48 (2007), 595–606.

    MathSciNet  MATH  Google Scholar 

  30. B. M. Vernikov, Lower-modular elements of the lattice of semigroup varieties, Semigroup Forum, 75 (2007), 554–566.

    Google Scholar 

  31. B. M. Vernikov, Lower-modular elements of the lattice of semigroup varieties. II, Acta Sci. Math. (Szeged), 74 (2008), 539–556.

    MathSciNet  MATH  Google Scholar 

  32. B. M. Vernikov, Upper-modular elements of the lattice of semigroup varieties, Algebra Universalis, 59 (2008), 405–428.

    MathSciNet  MATH  Google Scholar 

  33. B. M. Vernikov, Upper-modular elements of the lattice of semigroup varieties II, Fundamentalnaya i Prikladnaya Matematika (Fundamental and Applied Mathematics), 14:3 (2008), 43–51 (in Russian)

  34. B. M. Vernikov Engl. translation: J. Math. Sci., 164 (2010), 182–187.

    MathSciNet  Google Scholar 

  35. B. M. Vernikov, Codistributive elements of the lattice of semigroup varieties, Izvestiya VUZ. Matematika, 55 (2011), no. 7, 13–21 (in Russian)

    MathSciNet  MATH  Google Scholar 

  36. B. M. Vernikov Engl. translation: Russ. Math. Izv. Vuz, 55(2011), no. 7, 9–16.

    Google Scholar 

  37. B. M. Vernikov, Proofs of definability of some varieties and sets of varieties of semigroups, Semigroup Forum, 84 (2012), 374–392.

  38. B. M. Vernikov and V. Yu. Shaprynskĭi, Distributive elements of the lattice of semigroup varieties, Algebra i Logika, 49 (2010), 303–330 (in Russian)

    MathSciNet  MATH  Google Scholar 

  39. B. M. Vernikov and V. Yu. Shaprynskĭi Engl. translation: Algebra and Logic, 49 (2010), 201–220.

    MathSciNet  Google Scholar 

  40. B. M. Vernikov and M. V. Volkov, Complemented lattices of varieties and quasivarieties, Izvestiya VUZ. Matematika, 26 (1982), no. 11, 17–20 (in Russian)

    MATH  Google Scholar 

  41. B. M. Vernikov and M. V. Volkov, Engl. translation: Soviet Math. Izv. VUZ, 26 (1982), no. 11, 19–24.

    Google Scholar 

  42. B. M. Vernikov and M. V. Volkov, Lattices of nilpotent semigroup varieties, Algebraicheskie Systemy i Ih Mnogoobraziya (Algebraic Systems and their Varieties), L. N. Shevrin (ed.), Sverdlovsk: Ural State University, 1988, 53–65 (in Russian).

    Google Scholar 

  43. B.M. Vernikov and M. V. Volkov, Modular elements of the lattice of semigroup varieties II, Contrib. General Algebra, 17 (2006), 173–190.

    MathSciNet  MATH  Google Scholar 

  44. M. V. Volkov, Commutative semigroup varieties with distributive subvariety lattices, Contrib. General Algebra, 7 (1991), 351–359.

    MathSciNet  MATH  Google Scholar 

  45. M. V. Volkov, Semigroup varieties with modular subvariety lattices. III, Izvestiya VUZ. Matematika, 36 (1992), no. 8, 21–29 (in Russian)

    MathSciNet  Google Scholar 

  46. M. V. Volkov, Engl. translation: Russ. Math. Izv. VUZ, 36 (1992), no. 8, 18–25.

    Google Scholar 

  47. M. V. Volkov, Young diagrams and the structure of the lattice of overcommutative semigroup varieties, Transformation Semigroups, Proc. Int. Conf. held at the Univ. Essex, P. M. Huiggins (ed.), University of Essex, Colchester, 1994, 99–110.

    Google Scholar 

  48. M. V. Volkov, Modular elements of the lattice of semigroup varieties, Contrib. General Algebra, 16 (2005), 275–288.

    MathSciNet  MATH  Google Scholar 

  49. M. V. Volkov and T. A. Ershova, The lattice of varieties of semigroups with completely regular square, Monash Conf. on Semigroup Theory in Honour of G. B. Preston, T. E. Hall, P. R. Jones} and J. C. Meakin} (eds.), World Scientific, Singapore, 1991, 306–322.

    Google Scholar 

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Acknowledgements

The author thanks Drs. Olga Sapir and Edmond W. H. Lee for a number of useful suggestions on improvement of the initial version of this survey.

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Authors and Affiliations

  1. Ural Federal University, Institute of Mathematics and Computer Science, Lenina 51, Ekaterinburg, 620000, Russia

    B. M. Vernikov

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Communicated by M. B. Szendrei

Supported by the Ministry of Education and Science of the Russian Federation (project 2248), grant of the President of the Russian Federation for supporting of leading scientific schools of the Russian Federation (project 5161.2014.1) and by Russian Foundation for Basic Research (grant 14-01-00524).

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Vernikov, B.M. Special elements in lattices of semigroup varietie. ActaSci.Math. 81, 79–109 (2015). https://doi.org/10.14232/actasm-013-072-0

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