# Multiplicative Structure of Tropical Matrix Algebra

Lead Research Organisation:
University of Manchester

Department Name: Mathematics

### Abstract

Tropical algebra (also known as max-plus algebra) is the linear algebra of the real numbers when equipped with the binary operations of addition and maximum. It has applications in numerous areas of pure mathematics, applied mathematics, computer science and control engineering. An important aspect of tropical algebra is the algebraic structure of tropical matrices under multiplication, but so far there has been little systematic study of this topic. The research seeks to understand the abstract algebraic (semigroup-theoretic and group-theoretic) structure of the tropical matrices under multiplication, and in particular the semigroup of all n-by-n tropical matrices. As part of the research, we will implement a package for performing computations in tropical algebra using the GAP computer algebra system. Our research will also be informed by an examination of the possible applications.

## People |
## ORCID iD |

Mark Kambites (Principal Investigator) |

### Publications

Hollings C
(2012)

*Tropical matrix duality and Green's relation*in Journal of the London Mathematical Society
Izhakian Z
(2016)

*Pure dimension and projectivity of tropical polytopes*in Advances in Mathematics
Johnson M
(2013)

*Green's <mml:math altimg="si1.gif" display="inline" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="ht*in Journal of Pure and Applied Algebra
Kambites M
(2014)

*Idempotent tropical matrices and finite metric spaces*in Advances in Geometry
Wilding D
(2013)

*Exact rings and semirings*in Journal of AlgebraDescription | The research developed our understanding of the tropical semiring - a key mathematical structure with applications in many areas. It established new connections between the algebraic structure of tropical matrices under multiplication, and the geometry of tropical polytopes, contributing to understanding of both. |

Exploitation Route | This is an ongoing area of research, both within Manchester and beyond. |

Sectors | Chemicals,Digital/Communication/Information Technologies (including Software),Electronics,Manufacturing, including Industrial Biotechology,Transport,Other |

Description | This is "fundamental" research: although our work has clear potential for long-term impact in many areas, this is likely to take much longer to materialise. |