# Article

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Keywords:
max-min algebra; interval system; T6-vector; weak T6 solvability; strong T6 solvability; T7-vector; weak T7 solvability; strong T7 solvability
Summary:
Max-min algebra is an algebraic structure in which classical addition and multiplication are replaced by $\oplus$ and $\otimes$, where $a\oplus b=\max\{a,b\},\ a\otimes b=\min\{a,b\}$. The notation $\mathbf{A}\otimes \mathbf{x}=\mathbf{b}$ represents an interval system of linear equations, where $\mathbf{A}=[\underline{A},\overline{A}]$, $\mathbf{b}=[\underline{b},\overline{b}]$ are given interval matrix and interval vector, respectively, and a solution is from a given interval vector $\mathbf{x}=[\underline{x},\overline{x}]$. We define six types of solvability of max-min interval systems with bounded solution and give necessary and sufficient conditions for them.
References:
[1] A. Asse, P. Mangin, D. Witlaeys: Assisted diagnosis using fuzzy information. In: NAFIPS 2 Congress, Schenectudy 1983.
[2] K. Cechlárová: Solutions of interval systems in max-plus algebra. In: Proc. SOR 2001 (V. Rupnik, L. Zadnik-Stirn, S. Drobne, eds.), Preddvor 2001, pp. 321-326. MR 1861219
[3] K. Cechlárová, R. A. Cuninghame-Green: Interval systems of max-separable linear equations. Linear Algebra Appl. 340 (2002), 215-224. MR 1869429 | Zbl 1004.15009
[4] M. Gavalec, J. Plavka: Monotone interval eigenproblem in max-min algebra. Kybernetika 46 (2010), 3, 387-396. MR 2676076 | Zbl 1202.15013
[5] L. Hardouin, B. Cottenceau, M. Lhommeau, E. L. Corronc: Interval systems over idempotent semiring. Linear Algebra Appl. 431 (2009), 855-862. MR 2535557 | Zbl 1201.65070
[6] H. Myšková: Interval systems of max-separable linear equations. Linear Alebra. Appl. 403 (2005), 263-272. DOI 10.1016/j.laa.2005.02.011 | MR 2140286 | Zbl 1129.15003
[7] H. Myšková: Control solvability of interval systems of max-separable linear equations. Linear Algebra Appl. 416 (2006), 215-223. MR 2242726 | Zbl 1129.15003
[8] H. Myšková: An algorithm for testing T4 solvability of interval systems of linear equations in max-plus algebra. In: P. 28th Internat. Scientific Conference on Mathematical Methods in Economics, České Budějovice 2010, pp. 463-468.
[9] H. Myšková: The algorithm for testing solvability of max-plus interval systems. In: Proc. 28th Internat. Conference on Mathematical Methods in Economics, Jánska dolina 2011, accepted.
[10] H. Myšková: Interval solutions in max-plus algebra. In: Proc. 10th Internat. Conference APLIMAT, Bratislava 2011, pp. 143-150.
[11] A. Di Nola, S. Salvatore, W. Pedrycz, E. Sanchez: Fuzzy Relation Equations and Their Applications to Knowledge Engineering. Kluwer Academic Publishers, Dordrecht 1989. MR 1120025 | Zbl 0694.94025
[12] J. Rohn: Systems of interval linear equations and inequalities (rectangular case). Technical Report No. 875, Institute of Computer Science, Academy of Sciences of the Czech Republic 2002.
[13] E. Sanchez: Medical diagnosis and composite relations. In: Advances in Fuzzy Set Theory and Applications (M. M. Gupta, R. K. Ragade, and R. R. Yager, eds.), North-Holland, Amsterdam - New York 1979, pp. 437-444. MR 0558737
[14] T. Terano, Y. Tsukamoto: Failure diagnosis by using fuzzy logic. In: Proc. IEEE Conference on Decision Control, New Orleans 1977, pp. 1390-1395.
[15] L. A. Zadeh: Toward a theory of fuzzy systems. In: Aspects of Network and Systems Theory (R. E. Kalman and N. De Claris, eds.), Hold, Rinehart and Winston, New York 1971, pp. 209-245.

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