## Simple image set of (max,+) linear mappings.(English)Zbl 0976.15013

By defining an addition $$a\oplus b=\max(a,b)$$ and a multiplication $$a\otimes b =a+b$$ for any $$a,b\in\mathbb{R}$$, one can construct a max-algebra of real numbers. Then these operations can be extended to matrices and vectors in the same way as in conventional linear algebra. If $$A$$ is a real $$n\times n$$ matrix, then the mapping $$x\mapsto A\otimes x$$ from $$\mathbb{R}^n$$ to $$\mathbb{R}^n$$ ($$n>1$$) is neither surjective nor injective. However, for some of such mappings (called strongly regular) there is a nonempty subset (called the simple image set) of the range, each element of which has a unique pre-image. In this paper, a description of simple image sets is presented, from which criteria for strong regularity are obtained. It is also proved that the closure of the simple image set of a strongly regular mapping $$f$$ is the image of the $$k$$-th iterate of $$f$$ after normalization for any $$k\geq n-1$$ or, equivalently, the set of fixed points of $$f$$ after normalization.

### MSC:

 15A30 Algebraic systems of matrices 15A04 Linear transformations, semilinear transformations

### Keywords:

max-algebra; strong regularity; eigenproblem; assignment problem
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### References:

 [1] Baccelli, F.L.; Cohen, G.; Olsder J.-P. Quadrat, G.-J., Synchronization and linearity, (1992), Wiley New York [2] Butkovic, P., Strong regularity of matrices – a survey of results, Discrete appl. math., 48, 45-68, (1994) · Zbl 0804.06017 [3] Butkovic, P.; Cuninghame-Green, R.A., On the regularity of matrices in MIN algebra, Linear algebra appl., 145, 127-139, (1991) · Zbl 0731.15012 [4] Butkovic, P.; Hevery, F., A condition for the strong regularity of matrices in the minimax algebra, Discrete appl. math., 11, 209-222, (1985) · Zbl 0602.90136 [5] Cuninghame-Green, R.A., Describing industrial processes with interference and approximating their steady-state behaviour, Oper. res. quart., 13, 95-100, (1962) [6] Cuninghame-Green, R.A., Minimax algebra, Lecture notes in economics and mathematical systems, Vol. 166, (1979), Springer Berlin [7] R.A. Cuninghame-Green, Minimax Algebra and Applications in: Advances in Imaging and Electron Physics, Vol. 90, Academic Press, New York, 1995. [8] S. Gaubert, Théorie des systèmes linéaires dans les dioı̈des, thèse, Ecole des Mines de Paris, 1992. [9] M. Gavalec, Private communication, 1999. [10] M. Gondran, M. Minoux, Valeurs propres et vecteurs propres dans les dioı̈des et leur interprétation en théorie des graphes. EDF Bulletin de la Direction des Études et Recherches, Série C-Math. Info. #2, 1977, pp. 25-41. [11] Gondran, M.; Minoux, M., Linear algebra of dioı̈ds: a survey of recent results, Ann. discrete math., 19, 147-164, (1984) · Zbl 0568.08001 [12] Karp, R.M., A characterization of the minimum cycle Mean in a digraph, Discrete math., 23, 309-311, (1978) · Zbl 0386.05032 [13] Prou, J.M.; Wagneur, E., Controllability in the MAX-algebra, Kibernetika, 35, 13-24, (1999) · Zbl 1274.93036 [14] Wagneur, E., Moduloids and pseudomodules. I: dimension theory, Discrete math., 98, 57-73, (1991) · Zbl 0757.06008 [15] K. Zimmermann, Extremálnı́ algebra (Výzkumná publikace Ekonomicko-matematické laboratore pri Ekonomickém ústave CSAV, 46, Praha, 1976) (in Czech). [16] Zimmermann, U., Linear and combinatorial optimization in ordered algebraic structures, Annals of discrete mathematics, Vol. 10, (1981), North-Holland Amsterdam · Zbl 0466.90045
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