# AN OVERVIEW OF MAX-PLUS LINEAR SYSTEMS

Kim, Yong-Gu;Shin, Hyun-Hee

• Accepted : 2011.02.24
• Published : 2011.03.25
• 29 6

#### Abstract

Let $a{\oplus}b$ = max(a, b), $a{\otimes}b$=a+b, a, $b\in\mathbb{R}_{\varepsilon}\;:=\cup\{-\infty\}$. In max-plus algebra we work on the linear algebra structure for the pair of operations (${\oplus},{\otimes}$) extended to matrices and vectors over $\mathbb{R}_{\varepsilon}$. In this paper our main aim is to reproduce the work of R. A. Cuninghame-Green [3] on the linear systems over a max-plus semi-field $\mathbb{R}_{\varepsilon}$.

#### Keywords

Max-plus algebra;Linear equations;Tropical Geometry

#### References

1. F.L. Baccelli, G. Cohen, G.-J. Olsder, J.-P. Quadrat, Synchronization and Linearity, John Wiley, Chichester, New York, 1992.
2. P. Butkovic, Max-algebra: the linear algebra of combinatorics? Linear Algebra Appl. bf 367 (2003), 313-335. https://doi.org/10.1016/S0024-3795(02)00655-9
3. R. Cuninghame-Green, Minimax-Algebra, Lecture Notes in Economics and Math. Systems, Vol. 166, Springer-Verlag, New York, Heidelberg, Berlin 1979.
4. B. De Schutter, Max-Algebraic system theory for discrete event systems, Katholiceke Universiteit Press, 1994.
5. D. Speyer, B. Sturmfels, Tropical Matematics, Preprint arXiv:math.Co/04080099.