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3-Generated Lattice with Left Modular and Separating Generators

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Abstract

We consider a lattice generated by three elements, two of which are a left modular element and a separating element. It is proved that such a lattice is finite and contains at most 47 elements. It is shown that this estimate is accurate.

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References

  1. O. Ore, “On the foundations of abstract algebra-I,” Ann. of Math 36, 406–437 (1935).

    Article  MathSciNet  MATH  Google Scholar 

  2. G. Grätzer, “Standard ideals,” Magyar Tud. Akad. Mat. Fiz. Oszt. Közl 9, 81–97 (1959).

    MathSciNet  MATH  Google Scholar 

  3. G. Birkhoff, “Neutral elements in general lattices,” Bull. Amer. Math. Soc 46, 702–705 (1940).

    Article  MathSciNet  MATH  Google Scholar 

  4. G. Grätzer, Lattice Theory: Foundation (Springer Science & Business Media, 2011).

  5. G. Grätzer and E. T. Schmidt, “Standard ideals in lattices,” Acta Math. Acad. Sci. Hungar 12, 17–76 (1961).

    Article  MathSciNet  MATH  Google Scholar 

  6. A. G. Gein and M. P. Shushpanov, “Sufficient conditions for the modularity of the lattice generated by elements with properties of modular type,” Sibirsk. Mat. Zh. 56(4), 631–636 (2015) [Siberian Math. J. 56 (4), 798–804 (2015)].

    Article  MathSciNet  MATH  Google Scholar 

  7. A. G. Gein and M. P. Shushpanov, “Distributivity conditions of 3-generated lattices,” in International Algebraic Conference “Mal’tsev Readings” (Novosibirsk, 2014), pp. 131 [in Russian].

  8. N. A. Minigulov, On 3-generated lattices with standard and dual standard elements among generators, http://math.nsc.ru/conference/g2/g2s2/exptext/Minigulov-abstract-G2S2 (2016).

  9. S. P. Bhatta, “A characterization of neutral elements by the exclusion of sublattices,” Discrete Mathematics 309, 1691–1702 (2008).

    Article  MathSciNet  MATH  Google Scholar 

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Funding

This work was supported by the Project for Enhancing the Competitiveness (Agreement between the Russian Federation Ministry of Education and Science and Ural Federal University, August 27, 2013, no. 02. A03.21.0006).

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Authors and Affiliations

  1. Ural Federal University, Yekaterinburg, 620002, Russia

    M. P. Shushpanov

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  1. M. P. Shushpanov
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Correspondence to M. P. Shushpanov.

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The article was submitted by the author for the English version of the journal.

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Shushpanov, M.P. 3-Generated Lattice with Left Modular and Separating Generators. Math Notes 106, 113–117 (2019). https://doi.org/10.1134/S0001434619070125

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  • DOI: https://doi.org/10.1134/S0001434619070125

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