Tropical Arithmetic and Tropical Matrix Algebra
Zur Izhakian

TL;DR
This paper develops a generalized tropical semiring with modified arithmetic operations, exploring matrix invertibility and regularity within this new algebraic framework, bridging combinatorial structures with algebraic properties.
Contribution
It introduces a new commutative semiring extending tropical algebra, establishing conditions for matrix invertibility and regularity in this novel setting.
Findings
A new semiring structure generalizing tropical semiring
Invertibility of matrices characterized by regularity in the new framework
Connections between combinatorial properties and algebraic invertibility
Abstract
This paper introduces a new structure of commutative semiring, generalizing the tropical semiring, and having an arithmetic that modifies the standard tropical operations, i.e. summation and maximum. Although our framework is combinatorial, notions of regularity and invertibility arise naturally for matrices over this semiring; we show that a tropical matrix is invertible if and only if it is regular.
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 · Algebraic structures and combinatorial models
