# Tropical Arithmetic and Tropical Matrix Algebra

• Published in 2005
In the collection
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.

## Other information

arxivId
math/0505458
journal
ReCALL
keywords
Algebraic Geometry,Combinatorics
pages
17

### BibTeX entry

@article{Izhakian2005,
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.},
archivePrefix = {arXiv},
arxivId = {math/0505458},
author = {Izhakian, Zur},
eprint = {math/0505458},
journal = {ReCALL},
keywords = {Algebraic Geometry,Combinatorics},
month = {may},
pages = 17,
primaryClass = {math.AG},
title = {Tropical Arithmetic and Tropical Matrix Algebra},
url = {http://arxiv.org/abs/math/0505458 http://arxiv.org/pdf/math/0505458v3},
year = 2005,
urldate = {2012-02-03},
collections = {Unusual arithmetic}
}