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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:
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