Algebraic structures of tropical mathematics
Zur Izhakian, Manfred Knebusch, Louis Rowen

TL;DR
This paper introduces a layered algebraic framework for tropical mathematics that enhances the algebraic understanding of tropical geometry and linear algebra, extending beyond traditional max-plus algebra to include layered structures.
Contribution
It proposes a layered algebraic structure compatible with valuation theory, enabling a richer algebraic approach to tropical geometry and linear algebra, including new results on matrix rank and dependence.
Findings
Layered algebraic structures generalize max-plus algebra.
Tropical dependence characterized by ghost components in linear combinations.
Matrix rank related to the ghost nature of the permanent.
Abstract
Tropical mathematics often is defined over an ordered cancellative monoid , usually taken to be or . Although a rich theory has arisen from this viewpoint, cf. [L1], idempotent semirings possess a restricted algebraic structure theory, and also do not reflect certain valuation-theoretic properties, thereby forcing researchers to rely often on combinatoric techniques. In this paper we describe an alternative structure, more compatible with valuation theory, studied by the authors over the past few years, that permits fuller use of algebraic theory especially in understanding the underlying tropical geometry. The idempotent max-plus algebra of an ordered monoid is replaced by , where is a given indexing semiring (not necessarily with 0). In this case we say layered by . When is trivial, i.e, , is the usual…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsPolynomial and algebraic computation · Commutative Algebra and Its Applications
