Pure dimension and projectivity of tropical polytopes

Zur Izhakian; Marianne Johnson; Mark Kambites

doi:10.1016/j.aim.2016.08.033

Pure dimension and projectivity of tropical polytopes

Zur Izhakian, Marianne Johnson, Mark Kambites^*

^*Corresponding author for this work

Mathematical Science

University of Manchester

Research output: Contribution to journal › Article › peer-review

17 Citations (Scopus)

24 Downloads (Pure)

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.

Original language	English
Pages (from-to)	1236-1263
Number of pages	28
Journal	Advances in Mathematics
Volume	303
Early online date	13 Sept 2016
DOIs	https://doi.org/10.1016/j.aim.2016.08.033
Publication status	Published - 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.

Keywords

Max-plus algebra
Modules
Polytopes
Pure dimension
Rank
Tropical geometry

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Accepted Manuscript - IzhakianAdvances
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Cite this

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AU - Johnson, Marianne

AU - Kambites, Mark

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AB - 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.

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KW - Rank

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JF - Advances in Mathematics

ER -

Pure dimension and projectivity of tropical polytopes

Abstract

Bibliographical note

Keywords

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