Pure Dimension and Projectivity of Tropical Polytopes

Abstract

We study how geometric properties of tropical convex sets and polytopes, which are of interest in many application areas, manifest themselves in their algebraic structure as modules over the tropical semiring. Our main results establish a close connection between pure dimension of tropical convex sets, and projectivity (in the sense of ring theory). These results lead to a geometric understanding of idempotency for tropical matrices. As well as their direct interest, our results suggest that there is substantial scope to apply ideas and techniques from abstract algebra (in particular, ring theory) in tropical geometry.

Updated with author accepted manuscript, references improved, main results unchanged from previous version

References in corpus (3)

• Duality and separation theorems in idempotent semimodules
• Tropical polyhedra are equivalent to mean payoff games
• Tropical matrix duality and Green's D relation

Cited by in corpus (5)

• Identities in Upper Triangular Tropical Matrix Semigroups and the Bicyclic Monoid
• On congruence-semisimple semirings and the $K_0$-group characterization of ultramatricial algebras over semifields
• Idempotent tropical matrices and finite metric spaces
• Projective systemic modules
• The ultimate rank of tropical matrices