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Algebra i Analiz, 2014, Volume 26, Issue 2, Pages 216–228 (Mi aa1382)  

This article is cited in 1 scientific paper (total in 1 paper)

Research Papers

Tropical semimodules of dimension two

Ya. Shitov

National Research University Higher School of Economics, Myasnitskaya Ulitsa, 20, 101000, Moscow, Russia

Abstract: The tropical arithmetic operations on $\mathbb R$ are defined as $a\oplus b=\min\{a,b\}$ and $a\otimes b=a+b$. In the paper, the concept of a semimodule is discussed, which is rather ill-behaved in tropical mathematics. The semimodules $S\subset\mathbb R^n$ having topological dimension two are studied and it is shown that any such $S$ has a finite weak dimension not exceeding $n$. For a fixed $k$, a polynomial time algorithm is constructed that decides whether $S$ is contained in some tropical semimodule of weak dimension $k$ or not. This result provides a solution of a problem that has been open for eight years.

Keywords: tropical mathematics, linear algebra, computational complexity.

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English version:
St. Petersburg Mathematical Journal, 2015, 26:2, 341–350

Bibliographic databases:

Received: 27.06.2013
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Citation: Ya. Shitov, “Tropical semimodules of dimension two”, Algebra i Analiz, 26:2 (2014), 216–228; St. Petersburg Math. J., 26:2 (2015), 341–350

Citation in format AMSBIB
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\by Ya.~Shitov
\paper Tropical semimodules of dimension two
\jour Algebra i Analiz
\yr 2014
\vol 26
\issue 2
\pages 216--228
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3242042}
\elib{http://elibrary.ru/item.asp?id=21826356}
\transl
\jour St. Petersburg Math. J.
\yr 2015
\vol 26
\issue 2
\pages 341--350
\crossref{https://doi.org/10.1090/S1061-0022-2015-01341-1}
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\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84922289520}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Shitov Ya., “Extending orthogonal subsets of semimodules”, Linear Alg. Appl., 508 (2016), 225–233  crossref  mathscinet  zmath  isi  elib  scopus
  • Алгебра и анализ St. Petersburg Mathematical Journal
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