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Abstract
This article introduces a new structure of commutative semiring, generalizing the tropical semiring, and having an arithmetic that modifies the standard tropical operations, i.e., summation and maximum. Although our framework is combinatorial, notions of regularity and invertibility arise naturally for matrices over this semiring; we show that a tropical matrix is invertible if and only if it is regular.
2000 Mathematics Subject Classification:
ACKNOWLEDGMENTS
The author would like to thank Prof. Eugenii Shustin for his invaluable help. I'm deeply grateful to him for his support and the fertile discussions we had.
A part of this work was done during the author's stay at the Max Planck Institut für Mathematik (Bonn). The author is very grateful to Max Planck Institut for the hospitality and excellent work conditions.
The author has been supported by the Chateaubriand scientific post-doctorate fellowships, Ministry of Science, French Government, 2007–2008.
Notes
Communicated by M. Cohen.