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Weakly linear systems for matrices over the max-plus quantale

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Abstract

In this paper we introduce and study weakly linear systems, i.e. systems consisting of matrix inequalities Eqs. 17–20, over the max-plus quantale which is also known as complete max-plus algebra. We prove the existence of the greatest solution contained in a given matrix X0, and present a procedure for its computation. In the case of weakly linear systems consisting of finitely many matrix inequalities, when all finite elements of matrices X0, As and Bs, s ∈ I are integers, rationals or particular irrationals and a finite solution exists, the procedure finishes in a finite number of steps. If in that case an arbitrary finite solution is given, a lower bound on the number of computational steps is calculated. Otherwise, we use our algorithm to compute approximations to finite solutions.

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Notes

  1. we are talking about complete residuated lattices defined in the way that is the most common in the fuzzy set theory (cf. Ćirić et al. 2012a, b, 2010, Ignjatović and Ćirić 2012, Ignjatović et al. 2010, 2012, Jančić et al. 2016, Stamenković et al. 2018, 2014).

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Authors and Affiliations

  1. Department of Computer Science, Faculty of Sciences and Mathematics, University of Niš, Višegradska 33, 18000, Niš, Serbia

    Aleksandar Stamenković & Miroslav Ćirić

  2. Department of Mechanical Engineering, University of Texas, Austin, University Station C2200, Austin, TX. 78712-0292, USA

    Dragan Djurdjanović

Authors
  1. Aleksandar Stamenković
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  2. Miroslav Ćirić
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  3. Dragan Djurdjanović
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Correspondence to Aleksandar Stamenković.

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The first two authors are supported by Ministry of Education, Science and Technological Development, Republic of Serbia, Contract No. 451-03-68/2020-14/200124.

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Stamenković, A., Ćirić, M. & Djurdjanović, D. Weakly linear systems for matrices over the max-plus quantale. Discrete Event Dyn Syst 32, 1–25 (2022). https://doi.org/10.1007/s10626-021-00342-4

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  • DOI: https://doi.org/10.1007/s10626-021-00342-4

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