Tropical Arithmetic and Tropical Matrix Algebra

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.