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Lattices of Subalgebras of Semirings of Continuous Nonnegative Functions with the Max-Plus

An Erratum to this article was published on 09 February 2017

Abstract

Isomorphisms φ of semirings C (X) of continuous nonnegative functions over an arbitrary Hewitt space X with the condition φ( +) =  + are characterized in this work. It is proved that any isomorphism of lattices of all subalgebras of semirings C (X) and C (Y) is induced by a unique isomorphism of semirings excepting the case of one- and two-point Tychonovization of spaces.

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References

  1. 1.

    1. F. Cabello Sánchez, J. Cabello Sánchez, Z. Ercan, and S. Önal, “Memorandum on multiplicative bijections and order,” Semigroup Forum, 79, No. 1, 193–209 (2009).

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    R. Engelking, General Topology [Russian translation], Mir, Moscow (1986).

    Google Scholar 

  3. 3.

    I. M. Gelfand and A. N. Kolmogorov, “On rings of continuous functions on topological spaces,” Dokl. Akad. Nauk SSSR, 22, No. 1, 11–15 (1939).

    MATH  Google Scholar 

  4. 4.

    L. Gillman and M. Jerison, Rings of Continuous Functions, Springer, New York (1976).

    MATH  Google Scholar 

  5. 5.

    J. F. Golan, Semirings and Their Applications, Kluwer Academic, Dordrecht (1999).

    Book  MATH  Google Scholar 

  6. 6.

    6. G. Grätzer, General Lattice Theory [Russian translation], Mir, Moscow (1982).

    Google Scholar 

  7. 7.

    E. Hewitt, “Rings of real-valued continuous functions,” Trans. Amer. Math. Soc., 64, No. 1, 45–99 (1948).

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    J. Marovt, “Multiplicative bijections of C(X, I),” Proc. Amer. Math. Soc., 134, No. 4, 1065–1075 (2005).

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    E. M. Vechtomov, “Questions on the determination of topological spaces by algebraic systems of continuous functions,” Itogi Nauki Tekh. Ser. Algebra. Topol. Geom., 28, 3–46 (1990).

    MathSciNet  MATH  Google Scholar 

  10. 10.

    E. M. Vechtomov, “Rings of continuous functions. Algebraic aspects,” Itogi Nauki Tekh. Ser. Algebra. Topol. Geom., 29, 119–191 (1991).

    MathSciNet  MATH  Google Scholar 

  11. 11.

    E. M. Vechtomov, “Lattice of subalgebras of the ring of continuous functions and Hewitt spaces,” Mat. Zametki, 62, No. 5, 687–693 (1997).

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    E. M. Vechtomov, Introduction to Semirings [in Russian], VGPU, Kirov (2000).

    Google Scholar 

  13. 13.

    E. M. Vechtomov and V. V. Sidorov, “Isomorphisms of lattices of subalgebras of semirings of continuous nonnegative functions,” Fundam. Prikl. Mat., 16, No. 3, 63–103 (2010).

    MATH  Google Scholar 

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Correspondence to V. V. Sidorov.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 19, No. 6, pp. 153–189, 2014.

An erratum to this article is available at http://dx.doi.org/10.1007/s10958-017-3277-8.

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Sidorov, V.V. Lattices of Subalgebras of Semirings of Continuous Nonnegative Functions with the Max-Plus. J Math Sci 221, 409–435 (2017). https://doi.org/10.1007/s10958-017-3235-5

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