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.
Organisations
People |
ORCID iD |
Mark Kambites (Principal Investigator) |
Publications
Hollings C
(2010)
Tropical matrix duality and Green's D relation
Izhakian Z
(2011)
Pure Dimension and Projectivity of Tropical Polytopes
Hollings C
(2012)
Tropical matrix duality and Green's relation
in Journal of the London Mathematical Society
Wilding D
(2012)
Exact rings and semirings
Johnson M
(2013)
Green's J -order and the rank of tropical matrices
in Journal of Pure and Applied Algebra
Wilding D
(2013)
Exact rings and semirings
in Journal of Algebra
Kambites M
(2014)
Idempotent tropical matrices and finite metric spaces
in Advances in Geometry
Izhakian Z
(2016)
Pure dimension and projectivity of tropical polytopes
in Advances in Mathematics
Izhakian Z
(2017)
Tropical matrix groups
in Semigroup Forum
Description | 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. |