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Introduction

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Max-linear Systems: Theory and Algorithms

Part of the book series: Springer Monographs in Mathematics ((SMM))

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Abstract

Max-algebra is introduced as the linear algebra built upon linearly ordered commutative groups. The book is presented for the additive group of reals but the most typical alternative ground sets are listed. Following the essential definitions the concepts that play a key role in max-algebra are studied: the maximum cycle mean, transitive closures, conjugation and the assignment problem. This includes Karp’s algorithm for finding the maximum cycle mean and the Floyd-Warshall algorithm for finding the transitive closures. Essential properties of subeigenvectors are presented in this chapter; a detailed analysis of eigenvectors is postponed to Chap. 4.

Two types of problems that are of particular interest in this book, feasibility and reachability, are presented. They are related to the tasks of finding a steady regime in multi-machine interactive production processes and to synchronization and optimization of these processes.

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Notes

  1. 1.

    Except Sect. 1.4 and in the proof of Theorem 8.1.4.

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  1. School of Mathematics, University of Birmingham, Birmingham, UK

    Peter Butkovič

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Butkovič, P. (2010). Introduction. In: Max-linear Systems: Theory and Algorithms. Springer Monographs in Mathematics. Springer, London. https://doi.org/10.1007/978-1-84996-299-5_1

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