Pure dimension and projectivity of tropical polytopes

Zur Izhakian, Marianne Johnson, Mark Kambites*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

17 Citations (Scopus)
24 Downloads (Pure)


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.

Original languageEnglish
Pages (from-to)1236-1263
Number of pages28
JournalAdvances in Mathematics
Early online date13 Sept 2016
Publication statusPublished - 5 Nov 2016

Bibliographical note

The authors thank the anonymous referee for many useful comments, and in particular for drawing our attention to several related results in the literature.


  • Max-plus algebra
  • Modules
  • Polytopes
  • Pure dimension
  • Rank
  • Tropical geometry


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