Tropical Arithmetic and Tropical Matrix Algebra

Abstract

This paper 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.

17 pages, 2 figures

References in corpus (5)

• Tropical Geometry and its applications
• The Maslov dequantization, idempotent and tropical mathematics: A brief introduction
• Min-plus methods in eigenvalue perturbation theory and generalised Lidskii-Vishik-Ljusternik theorem
• Duality of Tropical Curves
• Tropical Linear Spaces

Cited by in corpus (5)

• Boolean Representations of Matroids and Lattices
• C-independence and c-rank of posets and lattices
• On tropical and Kapranov ranks of tropical matrices
• Cyclic classes and attraction cones in max algebra
• Superboolean rank and the size of the largest triangular submatrix of a random matrix