### Pure dimension and projectivity of tropical polytopes

Authors: Zur Izhakian, Marianne Johnson, Mark Kambites

Journal: Advances in Mathematics

Publication Date: 05 November, 2016

Department of: Mathematics

### In Abstract

**Exploring the relationships between geometric structure and algebraic structure**

Tropical mathematics (also known as max-plus mathematics) concerns the real numbers under the operations of addition and maximum. An active area of research since the 1970’s, it has well-documented applications in diverse areas including analysis of discrete event systems, optimisation and scheduling problems, formal languages and automata, and phylogenetics. Most recently, the discovery of deep connections with algebraic geometry has raised its profile within pure mathematics. A key role in many applications is played by tropically linear transformations: transformations of vectors whose descriptions appear linear once addition is recast as multiplication, and maximisation is recast as addition.

An important feature of such a transformation is its image: a mathematical object which has both a geometric structure (as a Euclidean polyhedral complex) and an algebraic structure (as a module over the tropical semiring). This research (conducted by researchers at the universities of Manchester and

Aberdeen) discusses the relationship between these geometric and algebraic structures. Specifically, a close correspondence between pure dimension (a property studied by geometers) and projectivity (a central notion in abstract algebra) is established. This leads to a geometric understanding of idempotency for tropical matrices, and also opens up new possibilities for applying techniques from abstract commutative algebra to problems in both tropical algebraic geometry and application areas.